Physics 2A Lab Manual

Transcription

1 Physics 2A Lab Manual Student Edition San José State University Spring 2016

2 Contents DL1 1 1 A Introduction to the Course Format B Pretest I Three-Phase Model of Matter & Energy-Interaction Model 3 DL2 5 2 A Making Sense of Thermal Phenomena B Analyzing a Heat Pack C Three-Phase Model of Matter & Energy-Interaction Model DL A Practicing Our Two Models B The Heat Pack and New Thermal Phenomena C Getting Quantitative: New Thermal Properties DL A Refining Intuitions through Practice DL A More Practice with the Energy-Interaction Model B Defining Appropriate Intervals for Analysis II Mechanical Energy Systems 35 DL A Modeling with Mechanical Energy Systems B A New Construct and A New Model DL A Creating Particular Models with Mechanical Energies B How We Use the Energy-Interaction Model C Quantitative Modeling of Mechanical Energies DL8 51

3 CONTENTS i 8 More Modeling with Mechanical Energy DL A Graphically Representing Energy Relationships B Explicit Reasoning Using Models DL A More Modeling of Mechanical Energy Systems B Connecting Potential Energy with Forces III Forces and Motion: Momentum Conservation 69 DL A Adding and Subtracting Vectors B Working with Position and Displacement Vectors DL A Check for Understanding: Vector Operations B Representing the Motion of a Mass along a Circle DL A Force Model: Net Force and Force Diagrams B Check for understanding: More Vectors DL A Momentum and Change in Momentum: 1-D Cases B Representing Momentum Conservation with Vectors C Check for understanding: 1-D Momentum DL A The Significance of t B Check for Understanding: The Egg Throw DL A 2D- and 3D-Momentum and Change in Momentum B Check for Understanding: Momentum IV Forces and Motion: Newton s Laws 103 DL A Newton s Laws and Momentum Conservation B Ball Swinging in a Horizontal Circle

4 Appendix A-1 SI Units and Conversions A-1 Three-Phase Model of Matter B-1 Energy-Interaction Model C-1 Energy in Mechanical Systems D-1 Momentum Conservation Model E-1 Newtonian Force Model F-1 List of FNTs Complete before DL3 13 FNT FNT FNT FNT Complete before DL4 23 FNT FNT FNT FNT FNT FNT FNT Complete before DL5 29 FNT FNT FNT FNT FNT FNT Complete before DL6 37 ii

5 LIST OF FNTS iii Complete before DL7 41 FNT FNT Complete before DL8 51 FNT FNT FNT FNT FNT FNT Complete before DL9 55 Complete before DL10 61 FNT FNT FNT FNT FNT FNT Complete before DL11 71 Complete before DL12 75 FNT FNT FNT FNT Complete before DL13 79 FNT FNT FNT FNT Complete before DL14 83 FNT FNT FNT Complete before DL15 89 FNT FNT FNT

6 iv LIST OF FNTS Complete before DL16 95 FNT FNT FNT FNT FNT FNT FNT FNT FNT FNT FNT FNT Complete before DL FNT FNT FNT

7 Discussion/Lab 1 1 A Introduction to the Course Format This class is run in a way that is likely different from what you are used to. Active participation is essential for your success in the course. To help you become a productive and successful member of our class community, we will start off with an introductory activity that will allow you to familiarize yourself with the course format. Since our philosophy about how you learn best in this course most likely differs from other instructors, we will do everything we can to help you get into the habit of participating actively and asking questions as soon as you have them. 1 B Pretest Our physics department is very interested in finding out about the effectiveness of our courses. Therefore, in many courses, students are asked to complete a pre-test at the beginning of the semester (and a post-test at the end) that aims to measure how well a particular course helps students understand certain physics topics. In the first discussion lab meeting, you will be taking a pretest to show what you know coming into the class. There is absolutely no need to study for this ahead of time. This test is not graded (however, we see taking the test as part of your active participation in this course). We may use the overall results to help us decide what to focus on teaching during the semester. 1

8 DL1 Activity 1 B. Pretest 2

9 Unit I The Three-Phase Model of Matter and The Energy-Interaction Model 3

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11 Discussion/Lab 2 2 A Making Sense of Thermal Phenomena Overview: We start our exploration of physics this semester by observing and reflecting on the phenomenon of freezing things. 2 A.1 Observing what happens when things freeze 1. Freezing a Heat Pack As a group, trigger one of the heat packs (you may know them as hand warmers ) on your table by clicking the button that s submerged in the liquid. Observe what happens within the heat pack during the first few minutes after triggering. Describe what you observe (i.e., tell each other a story about what happened to the heat pack). In a few sentences, write on the whiteboard your description (your story) of what you observed (not what you think happened!). Please use words that everyone in your group is comfortable with. Write clearly so that everybody in the room can read! 2. Freezing liquid water Based on your everyday experience, describe what you have to do to freeze water). After you have discussed this in your group for a few minutes, write your group s response on the board. 3. Does the behavior of a heat pack make sense? Discuss in your group: (a) Based on your everyday experience with freezing things, would you say that the heat pack froze when it was triggered? 5

12 DL2 Activity 2 A. Making Sense of Thermal Phenomena (b) Why or why not? (c) Does the behavior of the heat pack make sense to you? Why or why not? Write your thoughts in response to (a), (b), and (c) on the whiteboard. 2 A.2 Using models to make sense of physical phenomena: Introducing the Three-Phase Model of Matter Please read before moving on: Before we can make sense of the process that occurs when the heat pack is triggered, we need to organize what we know about freezing and melting. You probably studied the processes of freezing and boiling in several previous science classes in a physical science class or chemistry in high school and perhaps in a college chemistry class. You have also had a good deal of everyday experience with freezing and boiling. But it is likely that your knowledge about these processes and phases of matter is somewhat fragmented and not well organized. Pulling together what we know and getting it organized in a useful way is what we need to do. Learning Science means Refining Intuitions and Organizing Knowledge We all have everyday experience with lots of physical processes and phenomena. Sometimes, our intuitions seem to initially contradict the accepted scientific understanding. This is true for beginning scientists and experienced researchers alike! To continually improve our understanding of the world, we often need to refine our intuitions. In fact, we do this every day, while interacting with people, things, and situations in our environment. We use what we know about the world to make sense of our surroundings. Sometimes, we have to reorganize what we know in order to understand a new situation, and sometimes we learn something new altogether, new knowledge that we have to somehow fit in with what we already know. When this new knowledge organization becomes second nature to us, we have refined our intuition. Models help scientists organize knowledge One tool scientists use to organize knowledge related to a class of phenomena is the scientific model. Models bring together in one place the ideas and data patterns we use to make sense of phenomena in a concrete and useful way. A model represents these ideas in such a way that we can readily access them and use them. That is, a model typically provides the representational tools needed to use the model to develop explanations, make predictions, and more generally to make sense of phenomena. Over time, the features of a model will become second nature to us; we will have refined our intuitions about the phenomena in question. 6

13 Activity 2 A. Making Sense of Thermal Phenomena DL2 The Three-Phase Model of Matter The Three-Phase Model of Matter provides a summary of the ideas needed to make sense of phase changes in matter (really, it only applies to so-called pure substances, but we won t make that distinction yet). There are instances, however, when the standard model is not sufficient to understand certain phenomena that undergo a phase change. This is the case with the heat pack. We have to extend the model, which is what we will do together in the following activities. Temperature T Temperature vs. Energy-Added Diagram All liquid at T BP All gas at T BP gas phase T BP All solid at T MP All liquid at T MP liquid phase mixed phase: liquid/gas T MP solid phase mixed phase: solid/liquid Energy Added or Removed E (at constant pressure) 1. Examine the Three-Phase Model of Matter Summary There should be several copies of the model summary sheet on your table. You can also find this summary in the online resources. Take about five minutes to look at what is on the summary sheet. Then focus on the diagrammatic representation: The Temperature vs. Energy-Added Diagram. Discuss with your group what the axes mean. 2. Use the Three-Phase Model of Matter to construct a general explanation (a) Put a sketch of the complete Temperature vs. Energy-Added Diagram on your whiteboard that shows all three phases. Put a large dot on the graph that would represent tap water at room temperature. (b) Now with the whole group participating, have someone put their finger on the dot and the rest of the group (take turns) describing what is happening to the water (tell another story!) as it is brought to a boil in the electric hotpot on your table. The person with their finger on the dot should move her/his finger along the graph to match the description. Imagine you leave the heating pot on for five or ten minutes, but not so long that all the water boils away. Now imagine putting the pot in your freezer for an hour or so and move your finger accordingly along the cooling curve. (c) Repeat (b) over and over until everyone in your group is ready to explain to the whole class what goes on as you heat a substance way up and when you cool it way down. Use the language of the Three-Phase Model of Matter in your description, and use the diagram to illustrate your explanation. 7

14 DL2 Activity 2 A. Making Sense of Thermal Phenomena 3. Prepare the experiment Use the model (and your knowledge of freezing and boiling points of water) to make a prediction (i.e., don t start boiling the water yet!) about temperatures: (a) The hotpot in front of you should be filled about two thirds full with water. What temperature range does the model predict for the liquid water? I.e., what temperatures can liquid water have before it freezes or boils, according to the model? (b) Suppose you begin heating the water. Immediately after the water has begun to boil, what does the model predict for the temperature of the liquid water? What about after it has boiled for five minutes? Compare your initial intuition with what the model tells you the temperature absolutely must be (assuming the model is appropriate for this phenomenon). (c) If the lid is kept tightly on the pot and you have one thermometer immersed in the liquid and another one just above the surface of the liquid, and the water has been boiling vigorously for several minutes, what does the model tell you about the two temperatures? 4. Do the experiment (a) Check out the model s predictions with the hot pot and the two thermometers you have at your table. On the board make a chart and compare your predictions from part 3 with your experimental results. (b) Are there issues in the design of the experiment you just performed that are not taken into account in the model? Describe these on the board and be prepared to discuss them with the class. Remember: Make sure your responses are legible, and that your entire group agrees with what is written on the board. 5. What does the model say? Use the Three-Phase Model of Matter to answer the following question: When two phases of a substance are present and in thermal equilibrium (i.e., both phases have the same temperature), what do you know with certainty about certain properties or physical conditions as they relate to that substance? Discuss in your group. How does the model representation help you answer this question? 8

15 Activity 2 B. Analyzing a Heat Pack DL2 2 B Analyzing a Heat Pack Using the Three-Phase Model of Matter Overview: Our goal is to make sense of the heat pack behavior using both the Three-Phase Model of Matter and the Energy-Interaction Model. We ll start with a focus on the Three-Phase Model of Matter. 2 B.1 Macroscopic properties and energy transfers When analyzing a physical phenomenon, a model greatly simplifies and restricts what we focus on. The constructs that make up the model limit the aspects of the phenomenon that we must pay attention to. 1. Using only the constructs of the Three-Phase Model of Matter, what are macroscopic properties of the heat pack that changed (and how did they change) during the first few minutes after triggering? Look at the Model Summary Sheet and try to use only those words. Write your new story on the board. 2. How does the amount of energy associated with clicking the small disk inside the heat pack compare to the other energy transfers that occur? Think, for example, about the amount of heat transferred from the heat pack to your hands or to the environment. Is it about the same amount of energy? Much larger? Much smaller? Write your group s response on the board. 2 B.2 Completing the cycle Now, let s take the heat pack we had triggered before through the rest of its cycle. 1. Read and follow the instructions on the heat pack to get it back to being a liquid at room temperature. 2. Describe qualitatively and briefly how the properties of the heat pack identified in part 2 B.1 changed during the rest of the cycle. Write your new story on the board. Attention: You will continue your analysis of the heat pack in your homework assignment. Therefore, please write down in your notes how the properties changed as the heat pack went through its complete cycle. You will need this! 9

16 DL2 Activity 2 B. Analyzing a Heat Pack 2 B.3 Using our model 1. Think about your response to question 5 in Activity 2 A. 1 When your heat pack is in a mixed solid/liquid state, what do you know about its temperature? Use this knowledge to experimentally determine the melting-point temperature of the heat pack. In order to ensure equilibrium conditions, you can minimize energy transfers to or from the heat pack by wrapping plastic bubble-wrap around the heat pack and thermometer. You can also fold your heat pack around the end of the thermometer. Does it matter how you attained the mixed state: melting when in the hot pot or freezing when triggered? Record your result on the board, indicating a direction on your diagram. 2. Two processes that you can readily observe with the heat pack are: i) changing from liquid to solid and ii) changing from solid to liquid. (a) For which of these two processes does the heat pack seem to follow the predictions of the Three-Phase Model of Matter and for which process does it not follow the model? Be brief, but specific. Put your response on the board. [Hint: Use your finger to trace the graph.] (b) For which of these two processes does the heat pack behavior pretty much agree with your intuition about phase changes? For which process does it not agree? Put your response on the board. Does your intuition now agree pretty much with the model? If not, what is different? 1 Here s the question again, for reference, and so you don t need to flip back and forth: When two phases of a substance are present and in thermal equilibrium (i.e., both phases have the same temperature), what do you know with certainty about certain properties or physical conditions as they relate to that substance? 10

17 Activity 2 C. Three-Phase Model of Matter & Energy-Interaction Model DL2 2 C Practicing the Three-Phase Model of Matter; Introducing the Energy-Interaction Model Overview: We briefly interrupt our efforts of making sense of the heat pack to get a bit of practice using the Three-Phase Model of Matter and to introduce the Energy-System Diagram. Your instructor will assign one or more of the following physical processes to your small group: (a) Cooling a piece of solid copper (Cu) from 500 C to 350 C. (b) Warming a piece of ice from -20 C to the melting point. (c) Condensing steam completely to liquid at 100 C. (d) Completely sublimating a chunk of dry ice at -79 C. (e) Partially melting 25% of ice initially at 0 C. (f) Heating a piece of copper initially at 300 C until it is half melted. (g) Cooling and completely freezing H 2 O initially at 80 C. For each of these processes, sketch the following on your whiteboard and be prepared to explain to the whole class: (i) A complete Temperature vs. Energy-Added Diagram (remember, this is the graphical part of the Three-Phase Model of Matter) with the initial and final states clearly marked; and (ii) An open Energy-System Diagram. Refer to Steps Involved in Using the Energy-System Diagram on the Energy-Interaction Model summary on your table. Try your hand at translating your Energy-System Diagram into an algebraic expression of energy conservation. beginning Identification of beginning and end of interval: end initial conditions physical thing 1 physical system physical thing 1 physical thing 2 final conditions heat Q indicator a E 1,a a i = a f = indicator b E 1,b b i = a f = indicator a E 2,a a i = a f = ( ) (+) ( ) (+) E 1,a + E 1,b + E 2,a = Q 11

18 DL2 Activity 2 C. Three-Phase Model of Matter & Energy-Interaction Model Again, you need to be prepared to explain and describe the process you have been assigned to the whole class using both models. When you use the Three- Phase Model of Matter trace the process with your finger as you describe it. Make sure you clearly show the beginning and ending points of the process in the diagram. NOTE: You will have an opportunity to rework these examples in the homework. These are all very straightforward once you re familiar with the basic ideas of the two models. Working through these examples will also give you practice in applying the models and using the representations with a variety of phenomena. It is very important that you can do this quickly and correctly (i.e., it should become second nature to you). When applying these models to make sense of more complicated phenomena, e.g., on a quiz, it is crucial that you are not stumbling over the basics. 12

19 Discussion/Lab 3 3 A Practicing Our Two Models Overview: We continue to practice using the Three-Phase Model of Matter, and we ll start to get more familiar with the Energy System Diagrams of the Energy- Interaction Model. 3 A.1 Basics of the Three-Phase Model of Matter FNT 3-1 Scientists don t come up with right answers. Usually, when your instructor asks you to do problems for homework or on a test, they expect you to get a right answer, right? Well, we don t. As a matter of fact, for many of the problems you will encounter in this course, there won t be one right answer. Science just doesn t work that way. For every problem, there are a number of ways one can approach the problem, and each different solution path might result in different answers. But, that doesn t mean that one of them is right and the others are not. Scientists come up with arguments. Much more important than an answer to a question or problem in science is usually the method by which one arrived at this answer. So, our emphasis is on the method you use to solve a problem. The pathway to your solution. We want to know what assumptions and decisions you re making as you solve a problem, and how you re using these assumptions and decisions to make an argument for your solution. That s how science works! There is nobody who knows the right answer. But there are people who can look at what you ve done and who can tell you whether the assumptions you ve made are appropriate in a particular situation and the argument you ve constructed is valid and convincing. In science, this is called peer-review because the people who look at your work are other scientists. 13

20 DL3 Activity 3 A. Practicing Our Two Models So, if we don t care so much about you getting the right answer, how will we evaluate your progress in this course? Just like in a scientific publication, we ask you to be very specific about what you did and did not do, and how you arrived at a particular solution to a problem. We ll practice this a lot, so that you know what we expect from you. This first problem is an extreme example of what problem-solving in this class looks like for you: We ll actually give you some answers. A little further down, you ll find a box with some answers that one might reasonably get when solving this problem. Note that we re not saying these are the exact answers you will get or that we expect you to get! But your solutions to this problem might be quite similar to the ones given below. Now, if we give you the answers, what do we want you to do? We want you to describe in detail how you are using the diagram below to respond to the prompts in the problem. Write a story about the diagram and about how you are reading the graph to get the values you get. While this is stated a few times below, it s worth repeating: We don t want you to do any calculations for this problem! Instead, use the diagram and tell us how you re using the diagram and why we should believe you that your answer to the problem is a reasonable one. Yes, this might be a bit more work than what you re used to from other classes. Later in this course, we ll find shorter ways for you to show us your work and argument. But for now, please do write out a detailed story including your assumptions, decisions, etc. for each part of the problem. On to the actual problem: Use the particular Temperature vs. Energy-Added Diagram of the Three-Phase Model of Matter shown below to respond to the prompts in this FNT. The values along the x-axis (146 kj, 480 kj, etc.) are the amounts of energy that must be added to get the 1.0 kg of water from an initial temperature of 200 K to the next phase change. For example, 480 kj is the total amount of energy that must be added to the water starting at the initial temperature of 200 K to completely melt all the ice. 14

21 Activity 3 A. Practicing Our Two Models DL3 1. Describe what is happening at each of the corners on the graph. What phase is the water in at the letter A, B, C? 2. Assume that you have one kilogram of water at an initial temperature of 200 K. If 795 kj of energy is added, describe the final state (approximate temperature and phase) of the water. If the water is in a mixed phase, determine very approximately the percent in each phase. You are not expected to do any calculations at this time. Simply tell us what the diagram tells you (and how you know that it does). 3. Repeat Part 2 but with 146 kj of energy added to the water initially at 200 K. 4. Repeat Part 2 but with 1650 kj of energy added to the water initially at 200 K. 5. State the initial and final conditions of the process that takes the system from Point C to Point B and determine very approximately how much energy would have to be added or removed. 6. Repeat Part 5 for water initially in State A and ending in State C. Again: Do not make any calculations to respond to these prompts! You ve all done this FNT individually at home. Now you have an opportunity to practice creating scientific arguments together. Back up your claims with evidence and try to convince each other of your solution! 1. Discuss with your group and make sure every group member is confident that she/he can explain Question 1 on the FNT. 2. While there is not one single correct answer for Parts 2 6 of the FNT, you probably estimated the values somewhere in the vicinity of those in the box. Work out in your small groups any significant discrepancies you might have. EVERYONE in your group should be ready to explain how you obtained these results in the whole class discussion. Your instructor will tell your group which prompt you should respond to on the board. Possible Answers 2. liquid at 350 K 3. completely solid at 273 K 4. 1/3 gas, 2/3 liquid at 373 K 5. initial conditions: all gas at 373 K, final conditions: liquid at 50 C; E 2470 kj 6. initial conditions: 25% liquid at 0 C, final conditions: gas at 100 C; E 3000 kj 15

22 DL3 Activity 3 A. Practicing Our Two Models 3 A.2 Basics of the Energy-Interaction Model FNT 3-2 Neatly write out all of the Energy System Diagrams, with the accompanying Temperature vs. Energy-Added Diagrams, listed below (these are the same scenarios that were requested in Activity 2 C). Remember that complete Energy System Diagrams always include algebraic expressions of energy conservation. Refer to the Energy-Interaction Model discussion in the online resources. (a) Cooling a piece of solid copper (Cu) from 500 C to 350 C. (b) Warming a piece of ice from -20 C to the melting point. (c) Condensing steam completely to liquid at 100 C. (d) Completely sublimating a chunk of dry ice at -79 C. (e) Partially melting 25% of ice initially at 0 C. (f) Heating a piece of copper initially at 300 C until it is half melted. (g) Cooling and completely freezing H 2 O initially at 80 C. 1. Compare your responses for the processes in Activity 2 C, both the Temperature vs. Energy-Added Diagrams and the Energy System Diagrams. 2. Your instructor will tell you which one of the scenarios (a) through (g) to put on the board. Put the Temperature vs. Energy-Added Diagram and the Energy-System Diagram near each other. Make sure the two diagrams are consistent with each other and make sure everyone in the group can explain precisely why both are drawn the way you have them. 16

23 Activity 3 B. The Heat Pack and New Thermal Phenomena DL3 3 B Making Sense of the Heat Pack: Exploring New Physical Phenomena Overview: We return to our efforts of making sense of the heat pack by using both the Three-Phase Model of Matter and the Energy-Interaction Model. In the process, we ll explore some new, related thermal phenomena Super-Heating and Super-Cooling and we ll see that we have to modify the standard Three-Phase Model of Matter to describe and understand the heat pack. 3 B.1 Taking the Heat Pack through its Entire Cycle FNT 3-3 When using the Energy-Interaction Model to make sense of the behavior of a heat pack (or really, any thermal cycle), it helps to divide the overall process (cycle) into multiple sub-processes. Use the following sub-processes: Initial conditions Action Final Conditions (a) liquid at 100 C taken out of boiler liquid at 23 C (b) liquid at 23 C triggered solid/liquid at 54 C (and insulated) shortly after triggering (in mixed state) (c) solid/liquid at 54 C sitting on table solid at 23 C (d) solid at 23 C placed in boiler solid/liquid at 54 C (e) solid/liquid at 54 C left in boiler liquid at 100 C Make four Energy System Diagrams, one for each sub-process (a), (c), (d), and (e), but NOT (b). Also, make a Temperature vs. Energy-Added Diagram for each process. Make sure you can describe each process in your own words using both of the representations. Very important for the following: Make sure everyone in your group is ready to explain to the whole class how they know the answers to the following questions (detailed instructions below): 1. Does each energy system increase or decrease? 2. Is this an open or closed system with respect to energy, and if open, does energy come in or leave? 3. How are your responses consistent with your Temperature vs. Energy-Added Diagram? 4. And finally, how do both diagrams tell you about the sign of each term in the equation expressing energy conservation? 17

24 DL3 Activity 3 B. The Heat Pack and New Thermal Phenomena To make this task a bit more manageable, your instructor will assign one of the four heat pack processes listed in FNT 3-3 to your group: (a), (c), (d), or (e) note that we re not concerned with part (b) yet. For your assigned process, put on the board both (i) an Energy-System Diagram, and (ii) a Temperature vs. Energy-Added Diagram. Make sure that the two representations of the process are in agreement and that everyone in your group can explain why they are drawn the way they are. Refer to Steps Involved in Using the Energy-System Diagram on the model summary page. 3 B.2 The Clicking Mystery: Super-heating and super-cooling As we ll find out here, we have to modify the standard Three-Phase Model of Matter to illustrate what happens in the heat pack. Let s start with some preparations in this FNT. FNT 3-4 We want to analyze triggering step (b) in FNT 3-4 more closely using the Temperature vs. Energy-Added Diagram of the Three-Phase Model of Matter. This will help us get a deeper understanding of this part of the process and will enable us to extend the Three- Phase Model of Matter to make it more realistically explain actual phenomena. (It turns out that super cooling and super-heating are rather common. Usually, however, they occur over a very small temperature range and so go unnoticed.) 1. Sketch a Temperature vs. Energy-Added Diagram for sodium acetate between room temperature at approximately 150 C, assuming the sodium acetate does not undergo any super-cooling, i.e., assuming that the heat pack is solid at room temperature, and changes phase at its normal phase change temperature, which you determined in Activity 2 B. 2. On the same diagram, sketch the path representing the process of the liquid heat pack cooling down from 150 C to room temperature with no phase change (now assuming there is supercooling). Check that the path you sketched makes sense by verifying that the changes in temperature and energy shown on the diagram as you move along the path you sketched match what you know about the actual changes in temperature and energy of the heat pack as it cools to room temperature without changing phase from liquid to solid. 1. Extending the Temperature vs. Energy-Added Diagram Now you are going to take the two curves from FNT 3-4, and draw them on top of each other. The trick is to figure out where they exactly overlap and where the two curves deviate from each other. 18

25 Activity 3 B. The Heat Pack and New Thermal Phenomena DL3 Erase your board and draw the Temperature vs. Energy-Added Diagram fairly large that you drew for process 2 in FNT 3-4: Cooling from 100 C to 23 C with no phase change. Now, we are going to draw the combined heating process (d) plus (e) in the same diagram. BUT WAIT! The heating curve you draw will need to be on top of one another where they describe the exact same state: namely warming or cooling the liquid from 54 C to 100 C. They are in the same phase, liquid, when they are at 100 C (and consequently in the same state), but they are in different phases when they are at 23 C, so they are not in the same state. When they are in different states, the curves won t be on top of each other. Make sure your heating curve and cooling curve overlap where they are supposed to. Make sure everyone in your group knows why the two curves overlap the way they do. 2. What happens when the clicker is clicked? Now, to make the analysis a little simpler, let s assume this: When you activate your heat pack, it is thermally insulated like it is when wrapped tightly with bubble wrap. Sketch the path of the clicking process on the cooling curve you just made above starting at the bottom end of the cooling curve at 23 C. You can determine this new path after the click from what you know from your experiments about the temperature change, and from the implications of heat-in or heat-out because it was insulated with the bubble wrap. What state (temperature and phase) is the sodium acetate in shortly after the clicking has occurred and while still insulated? Use both what you know from your direct observation of the phenomenon and from your diagrams to draw this portion of the process (from 23 C to 54 C). Make sure the new part of the path you draw on the diagram (from before to shortly after clicking) describes this process. 3. The Energy-System Diagram for clicking Now make an Energy-System Diagram for the clicking process. Open or closed? Which energy systems? Which energy system increased and which decreased? What feature of your Energy-System Diagram tells you that the part of the graph in the Temperature vs. Energy-Added Diagram that represents the clicking is simply a vertical straight line? 4. Getting back to room temperature If you now partially remove the bubble wrap so the heat pack can exchange energy with the environment, sketch the rest of the path that gets the heat pack back to a solid at room temperature. Should this path fall on top of the heating path? 19

26 DL3 Activity 3 B. The Heat Pack and New Thermal Phenomena 5. Three-Phase Model of Matter with Super-Cooling Discuss in your small group how you can summarize the difference in the Three-Phase Model of Matter as applied to situations when there is and when there isn t significant/noticeable supercooling. Put your summary on your board and be ready to explain to the whole class. 20

27 Activity 3 C. Getting Quantitative: New Thermal Properties DL3 3 C Getting Quantitative: New Thermal Properties Overview: So far, we ve used qualitative descriptions and we ve estimated numbers to describe the thermal phenomena occurring in the heat pack. However, we often need to be able to accurately quantify our observations and come up with mathematical models that can hopefully give us precise predictions for measurements. In this section, we ll familiarize ourselves with some mathematical models for thermal properties. In particular, we ll talk about Heat Capacity and Specific Heat, as well as Heat of Vaporization and Heat of Melting. Your instructor will give you one or two minutes to discuss each of the following in the following parts 3 C.1 and 3 C.2 in your Small Groups. Quickly come to a consensus and put your response on the board. Each question will be followed by a brief. 3 C.1 Heat Capacity 1. You may have used the relationship Q = C T in chemistry and physical science classes. What does heat capacity mean in your own words? You don t have to put this on the board, but make sure everyone in your group is ready to explain. 2. Imagine you have a big bucket of water and a small (like, European-sized) teacup of water, which would have the greater heat capacity the water in the bucket or in the teacup? Which would have the greater specific heat? 3. Create an open-system Energy-System Diagram that illustrates the process of making a heat capacity measurement for the big bucket of water (adding heat to this water). Note: We are assuming that this measurement takes place at a constant volume (no water is lost during the measurement). The significance of this will make more sense later in the course. 4. Use your diagram from (3) and the definition of heat capacity to develop an algebraic relationship that relates change in thermal energy to temperature change and heat capacity. 5. Use the following definitions to determine simple algebraic relationships between the heat capacity and each of two specific heats (mass & molar): Mass specific heat is the heat capacity per kilogram, and Molar specific heat is the heat capacity per mole. In what situations would one or the other be more useful or convenient to use? 21

28 DL3 Activity 3 C. Getting Quantitative: New Thermal Properties 3 C.2 Heat of Vaporization and Heat of Melting 1. You may have used the relationship Q = H in chemistry and physical science classes. What does heat of melting or heat of vaporization mean in your own words? As above, you don t have to put this on the board, but you should make sure everyone in your group is ready to explain. 2. Use the Energy-Interaction Model to illustrate a process in which Q = E BOND. 3. If H were given in units of joules per kilogram ( J /kg) or units of joules per mole ( J /mol), instead of units of joules, how would the equation, Q = H, change? 4. Use questions 1 through 3 above to describe a specific relationship between the change in bond energy in our Energy-Interaction Model, and the indicator of the bond energy system (i.e., the portion of mass of a substance that changed phase). 5. Rewrite the equation expressing energy conservation for process (b) in FNT 3-3, using the expressions for E th and E BOND that were introduced in this activity in terms of C and H. Then substitute all known values for each variable in your expression. Let s think about our heat pack again: What information do you need to determine how much mass will change phase when the button is clicked, assuming the heat pack is completely thermally insulated? 22

29 Discussion/Lab 4 4 A Refining Intuitions through Practice Overview: Remember how we talked about refining intuitions and becoming so familiar with using our models that they become second nature to us? For that to happen, we need to practice using our models even more. This is what the following activities are all about. We ll attempt to make sense of more thermal phenomena by using the Three-Phase Model of Matter and the Energy-Interaction Model, so that we ll become able to intuitively use them in new situations. 4 A.1 Energy in ice and liquid water FNT 4-1 Use the Energy-Interaction Model to explain whether the following statement is true or false. A quantity of ice at 0 C must contain less total energy than the same quantity of water at 0 C. Make an Energy-System Diagram that relates the initial and final states described in the FNT. Use the diagram, and the logic of the model to explain which state would have more total energy. 4 A.2 The physical concept of Heat FNT 4-2 According to the definition of heat in your course notes (pages 8 & 16), can an energy system contain a certain amount of heat? Explain. Discuss your response for this FNT with your group mates. Make sure everyone in your group is prepared to explain to the whole class. This is a group responsibility! 23

30 DL4 Activity 4 A. Refining Intuitions through Practice 4 A.3 Equilibrating Copper and Water FNT 4-3 Imagine that you place a piece of copper with an initial temperature of 20 C in contact with an amount of liquid water with an initial temperature of 100 C. Assume that the physical system consisting of the copper and the water is thermally isolated from everything else; i.e., water and copper can only exchange energy with each other. (a) Using the Three-Phase Model of Matter as applied to each substance and your understanding of what coming to thermal equilibrium means: i. Sketch two Temperature vs. Energy-Added Diagrams one for each substance and indicate the initial state for each one. ii. Explain in a sentence or two how you can tell if either substance will undergo a phase change during the process of coming to thermal equilibrium. You are expected to use what you already know regarding the thermal properties of copper and water, but you do not need to do any calculations. (b) Construct a complete Energy-System Diagram for the process that ends when the two substances are in thermal equilibrium. Don t forget that a complete diagram always includes an algebraic expression of energy conservation. (c) Now consider a similar process. Use the same initial conditions for the copper, but assume that the H 2 O is initially in the gas phase at 100 C. i. Sketch two Temperature vs. Energy-Added Diagrams, one for each substance, and mark the initial state for each one. ii. How can you determine if the H 2 O will undergo a temperature change? (What calculations and comparisons think energy would you need to make? Don t actually do the calculations; just explain what you would need to do.) 1. Put your Temperature vs. Energy-Added Diagrams on the board with the initial state marked on each one. Explain briefly how you can tell if either substance might undergo a phase change during this process. 2. Put your complete Energy-System Diagram on the board for this process. If E th(cu) = 250 kj, what is E th(h2 O)? Explain why. 3. Redraw your Temperature vs. Energy-Added Diagrams for the new initial conditions. Explain briefly how you could determine if the H 2 O will undergo a phase change or temperature change. 24

31 Activity 4 A. Refining Intuitions through Practice DL4 4 A.4 Thermal and Bond Energy Systems FNT 4-4 Consider again the phenomenon of two substances at different temperatures exchanging energy as they come to thermal equilibrium: Substance A and Substance B, at initially different temperatures, are placed in contact. They are able to exchange energy as heat, and they can only exchange it with each other. Substance A has a greater initial temperature than Substance B. No phase changes occur during this process. Use the Energy-Interaction Model to create a logical explanation that unequivocally shows that the following statement about the heat exchange described above for Substances A and B is true or that it is false: The energy that is transferred as heat to or from the object with the larger heat capacity must be greater than the energy that is transferred as heat to or from the object with the smaller heat capacity. Discuss this FNT in your group and be prepared to explain in the whole class discussion. You do not need to put anything on the board. FNT 4-5 Consider the following example of two substances at different temperatures exchanging heat as they come to thermal equilibrium. A 3.50 kg piece of lead and a 1.5 kg piece of aluminum, at different temperatures, are placed in contact. They are able to exchange energy as heat and they can only exchange it with each other. The initial temperature of the lead is 48 C, and the initial temperature of the aluminum is 35 C. The table on page 5 the course notes has phase change temperatures and values of specific heats for these substances. (a) If it is possible (How do you know this?), construct a single, closed Energy-System Diagram for the process of lead and aluminum coming to thermal equilibrium. (b) Write an algebraic expression of energy conservation in terms of the Es of the two energy systems, and then substitute in the appropriate expressions for the Es. (c) Substitute in all known numerical values. How many unknowns are there in your equation expressing energy conservation? (d) At an intermediate point in the entire process described above the lead has reached a temperature of 43 C. Follow the instructions for part (a) for the interval that ends when the lead is at a temperature of 43 C. What feature related to the process connects the final temperatures of lead and aluminum that appear in your diagram to each other? 25

32 DL4 Activity 4 A. Refining Intuitions through Practice 1. A student constructed the following Energy-System Diagram for part (a) of this FNT: Discuss the diagram briefly in your group, and identify any questions you have. You don t need to put anything on the board Put a complete Energy-System Diagram on the board for the shorter interval described in part (d) of this FNT. What specifically is happening in the process that connects the two final temperatures (of lead and aluminum) in your diagram? Illustrate this on the board with two separate Temperature vs. Energy-Added Diagrams, one for each substance. FNT 4-6 Suppose you used a hot pot to convert a 150 g piece of ice that was initially at -15 C into liquid water at 50 C. (a) Represent this process in two complete Energy System Diagrams, one covering the interval from -15 C to 0 C liquid, and the second covering the interval from 0 C liquid to 50 C. (b) Now represent the entire process in one complete Energy-System Diagram covering the entire interval. (c) If your hot pot has a power rating of 600 watts, show how to find how long it will take to complete the process. (Make sure you are comfortable using standard units of energy, and that you understand the relation of power to energy.) 1. Again, we consider a student s response to this FNT. Discuss the diagrams for part (a) below and identify any questions you have. As before, you don t have to put anything on the board for this part. 1 Incidentally, this diagram illustrates how much algebra is useful to put down for the purpose of clear communication in whole class discussions. In general, it is not useful to show the details of solving the algebra on the board. Being able to construct the correct diagram and getting the first three lines of algebra (including verifying the signs of the terms in the third line) are worth most of the credit on an exam. 26

33 Activity 4 A. Refining Intuitions through Practice DL4 2. A student objects to the diagram for part (b) of this FNT below because some of the indicators do not correspond to the beginning and end of the overall interval. Another student says that s ok because each physical process in the interval happens sequentially and nothing has been left out. Come to a consensus about which of these two students you agree with and put a brief explanation on the board. 3. Discuss the question about power in part (c) in your group and identify any questions you have. Make sure you have a clear understanding of the relationship of power to energy. You don t have to put anything on the board for part (c), but be ready to explain the difference between energy and power if called upon in the whole class discussion. 4 A.5 Asking Questions and Determining Intervals of Interest FNT 4-7 FNT 4-5 and FNT 4-6 emphasized the significance of the beginning and end of the interval we ve called them initial and final states so far. This FNT illustrates (i) that the question you re asking determines the interval being analyzed with the Energy-Interaction Model; and 27

34 DL4 Activity 4 A. Refining Intuitions through Practice (ii) that sometimes it s necessary to adjust the size of the interval in order to be able to use the model. The physical process we want to analyze with the Three-Phase Model of Matter and the Energy-Interaction Model is the following: Imagine you are using your hot pot to gradually add heat to a 500 g block of ice that has just been removed from a freezer with an internal temperature of -25 C. The hot pot is fairly well insulated, so it is reasonable to assume that all of the heat put into the pot from the electrical heater located in the bottom of the pot goes into the H 2 O. We can also assume that the heat capacity of the pot is much smaller than the heat capacity of 500 g of H 2 O (whether solid or liquid), so we can ignore the thermal energy system of the pot itself. One question we could ask is, What is the final state of the water (phase and temperature) after the addition of 252 kj of heat? (a) Explain why based only on the given information and without further analysis or calculation it is not possible to construct a single Energy-System Diagram that could be used to determine the final state of the water. [Hint: try to construct such a diagram. Another way to think about this is whether you can define the final state so that it depends on only one variable without making unjustified assumptions?] (b) In cases like this, you must first use the model to analyze a shorter interval. Can you construct an Energy-System Diagram that could be used to determine how much energy is needed to increase the temperature of the ice to 0 C? Now explain in a few sentences how to proceed using the Energy-Interaction Model to find the final state of the H 2 O. 1. Why is it not possible to apply the Energy-Interaction Model in a straightforward way to the interval of interest the interval that ends when all of the 252 kj has been added to the water? In other words, why can t the overall interval be diagrammed using only the given information prior to doing any further analysis? Draw a Temperature vs. Energy-Added Diagram to help you make sense of this and to use to explain it. In particular, be sure to make explicit which part of the graph in the diagram refers to which energy bubble on the Energy-System Diagram. [To save time, it is ok for explanations on the board to be more abbreviated than explanations on exams because anything that is unclear can be easily clarified in the whole class discussion.] 2. Explain how to use the Energy-Interaction Model to analyze shorter intervals in order to determine the final state (phase and temperature) of the water. (Note: water, without a modifier, usually means H 2 O in any one of its Three-Phases.) Use a Temperature vs. Energy-Added Diagram in your analysis and to use in your explanation. 28

35 Discussion/Lab 5 5 A More Practice with the Energy-Interaction Model Overview: We continue our efforts to practice using our models particularly the Energy-Interaction Model by investigating new phenomena. 5 A.1 Cooling Water Below its Freezing Point FNT 5-1 Perhaps you recall that when table salt, NaCl, is added to water, the freezing point of water is lowered. Consider a system composed of a mixture of 2.5 kg of ice and 50 g of liquid water and a small, separate container of finely powdered salt. This physical system is contained in a fully insulated container that prevents all thermal interactions with the environment. Both the salt and the ice-water mixture are initially at the freezing point of water, 0 C. The salt is then added to the ice-water mixture, and the system of ice-water and salt is allowed to come to thermal equilibrium. The final equilibrium temperature is less than 0 C. Use the Energy-Interaction Model to predict if there will be a greater or lesser amount of ice in the final equilibrium state than in the initial state before the salt was added. Your explanation should include a complete Energy-System Diagram. [The simplest way to model this physical system is with one thermal energy system for everything and one bond energy system, i.e., in terms of the model, it is not useful to distinguish between the various chemical components in order to answer this particular question.] On the board, construct the simplest Energy-System Diagram that represents a particular model of the process described in this FNT. Make sure everyone in your group is ready to explain which elements of your completed diagram represent information that was given, or known, and which elements you had to infer, based on given or known information and the logic of the model? 29

36 DL5 Activity 5 A. More Practice with the Energy-Interaction Model FNT 5-2 This FNT is a thought experiment to make predictions about an actual experiment we will do in class, where you will get to observe the phenomenon described in FNT 5-1. Imagine you fill an insulated cup almost full with chopped or crushed ice, and measure the temperature after a minute or two, once it s all come to thermal equilibrium. Since this ice is frozen water, the temperature should be at 0 C or not more than a couple of degrees below. Then, imagine you add a bunch of salt and stir it around. What do you think the lowest temperature you can attain will be? Why? What happens to the amount of liquid present if your keep stirring and adding salt? How can we understand this phenomenon in terms of thermal and bond energy systems? Develop an explanation for the changes you would observe in this physical system (decrease in temperature and change of phase) in terms of the Energy-Interaction Model. Now we ll actually do this experiment: 1. On your table, you have an insulated mug and a thermometer. Get some ice from the ice chest on the counter and fill your insulated mug. Measure the temperature inside the ice-filled cup and verify it is at around 0 C. 2. Now, add some salt and stir it into the ice with a stirrer (don t use the thermometer; it might break!). Measure the temperature of the salt-ice mixture inside the cup. Add some more salt and repeat the stirring and temperature measurement. How far can you get the temperature to drop? 3. On the board, list the low temperatures attained by your group. Would the following statement be something that would be good to memorize? When bond energy increases, thermal energy must decrease, and vice versa. Explain on the board why it would or would not be good to memorize this. 5 A.2 Heating Water with Microwaves FNT 5-3 This is an actual experiment that you should perform at home: Put approximately 1 cup of cold water in a microwave safe dish (Tupperware, drinking glass, etc.), measure the temperature (you can use a thermometer you would use to measure your body temperature), then heat it for a set period of time (somewhere between 30 seconds and a minute will work well). When you take the water out, measure the temperature again. 30

37 Activity 5 A. More Practice with the Energy-Interaction Model DL5 (a) Draw an Energy-System Diagram for the water in your cup, and determine the change in thermal energy of the water. Convert this to Watts (a unit of power). (b) Compare the number of Watts you determined in part a) to the power rating listed on the microwave information plate (usually located on the back of the oven, but you can always look up the model online). Which is greater? What does this mean? (c) Where might the energy go if it is not heating up the food? (d) Does the percentage of energy going to the food depend on the amount of water? You might choose a small amount, say half a cup, and a large amount, say two cups to investigate this question. On the board, choose the data from a member of your group, and sketch a graph of cooking power vs. amount of water. You ll notice that there appears to be some missing power. That is, the difference in power actually used to heat the water is less than the power used by the unit as stated on a label on the back of the microwave unit. [Hint: Can you feel or hear anything near the back of the microwave unit when it is turned on?] 5 A.3 Boiling Liquid Nitrogen with Ice FNT 5-4 A 2.2 kg block of ice (H 2 O) initially at a temperature of -20 C is immersed in a very large amount of liquid nitrogen (N 2 ) at a temperature of -196 C. The N 2 and H 2 O are allowed to come to thermal equilibrium. [T BP(N2) = -196 C] Create a particular model of this process and use it to determine how much liquid N 2 is converted to gas (vapor). [Hint: The emphasis on very large implies that there will still be liquid N 2 left when the two come to thermal equilibrium.] 1. There exits a definite correlation between each energy system in your Energy-System Diagram and each E term in your energy conservation equation. In addition to ensuring that you include all the appropriate E s (and no extra ones) in your algebraic equation expressing conservation of energy, what other very important detail can your Energy-System Diagram tell you about each E term in your equation? 2. Construct a complete Energy-System Diagram (including an algebraic statement of energy conservation in terms of the E s) for this process. Then rewrite the equation expressing energy conservation substituting in the appropriate expressions for changes in energy of the energy systems using variables only (no numbers). Do not substitute any numbers for the variables, and do not solve for the unknown, but circle 31

38 DL5 Activity 5 A. More Practice with the Energy-Interaction Model the unknown variable in the equation. Put the algebraic sign in parentheses above each algebraic term in the algebraic equation. We know you know how to do simple algebra and arithmetic. However, the most common mistake students often make is in the algebraic sign of the whole term when subtracting initial values from final values of the indicators. By using the Energy- System Diagram to check that the sign is correct, you will avoid making this error. 1 5 A.4 Warming Ice with Liquid Water FNT 5-5 The 2.2 kg block of ice from the FNT 5-4 is eventually removed from the liquid N 2 (after reaching thermal equilibrium with the liquid N 2 ) and placed in a very large amount of liquid H 2 O at 0 C, where it comes to thermal equilibrium with the liquid H 2 O. Create a particular model of this process and use it to determine the final mass of the ice. 1. Describe in words what is going on in this process, specifically: What physical system is changing phase? What system is changing temperature? What energy systems need to be included in the particular model you create for this process? 2. Construct a complete Energy-System Diagram (including an algebraic statement of energy conservation in terms of the E s) for this process. Then rewrite the equation that expresses energy conservation, substituting in the appropriate expressions for changes in energy of the energy systems using variables only (no numbers). Do not substitute any numbers for the variables, and do not solve for the unknown, but circle the unknown variable in the equation. 2 5 A.5 Melting Gold FNT kg of solid gold (Au) at an initial temperature of 1000 K is allowed to exchange heat with 1.5 kg of liquid gold at an initial temperature of 1336 K. The solid and liquid can only exchange heat with each other. When the two reach thermal equilibrium, will the mixture be entirely solid, or will there be a mixed solid/liquid phase? Explain how you know. Discuss in your group and make sure everyone is prepared to explain your group s response in the whole class discussion. 1 Just for reference: You likely found that 4 kg of liquid N 2 was converted to vapor. 2 Again, for reference: You likely got a final mass of ice of 5 kg. 32

39 Activity 5 B. Defining Appropriate Intervals for Analysis DL5 5 B Defining Appropriate Intervals for Analysis Overview: Remember that we talked about how asking different questions makes us choose different intervals of interest for our investigations using the Energy- Interaction Model? Here, we ll explore this issue a bit further. Consider the following phenomenon: A big chunk of ice (water) at an initial temperature of 65 C is placed inside a well-insulated container with some tea at an initial temperature of 20 C, and the two are allowed to come to thermal equilibrium. (The tea can be treated as water with respect to thermal properties.) 1. There are several possible final states of this process. We will be using and reusing the same Three-Phase Model of Matter to determine possible final states. Draw two Temperature vs. Energy-Added Diagrams, one for tea and one for H 2 O. (a) Why can t either substance reach thermal equilibrium in the gaseous phase nor the mixed (liquid/gas) phase nor the gaseous phase? (b) Using the grid below, your instructor will assign your group one of the potentially possible final states. Trace your Temperature vs. Energy-Added Diagram to show how your assigned ending phase is or is not possible. Table 5.1: Grid for 1b Tea Mixed Phase Solid (Solid/Liquid) i. ii. iii. Liquid Solid iv. v. vi. Ice Mixed Phase (Solid/Liquid) vii. viii. ix. Liquid 33

40 DL5 Activity 5 B. Defining Appropriate Intervals for Analysis 2. Now that we have a class consensus on which states are possible, your instructor will assign you one of the possible final states: (a) For your possible final state, draw an Energy-System Diagram (with energy conservation equation, as always). (b) Would your final state be possible if there were A LOT of ice? What about with A LOT of tea? 3. Thinking about how to proceed: How do we determine the final state? Put a Temperature vs. Energy-Added Diagram on the board for each of the two systems: The water that starts out as ice at -65 C and the tea (liquid water) that starts out at 20 C. Put a big dot on each graph at the initial state. (a) Have two separate group members place a finger on the two starting dots. Then a third group member should start explaining how energy is transferred from one physical system to the other, specifically naming the energy systems that are losing energy and that are gaining energy. Move the fingers along each graph in response to the explanation. Make sure every group member agrees with the explanation and the motion along the graphs. STOP when one substance gets to a phase change temperature. (b) We need to use quantitative skills to figure out which substance gets to its phase change temperature first. With your group members, think about ways that you might be able to determine this. Hint: Are there two values you can compare? Write in words what you need to do to figure this out. (c) After you have explained in words how you will figure out which substance gets to its phase change first, draw an Energy-System Diagramfor each calculation you want to perform, and then perform the calculation. (d) Is it possible to determine the final state of the substances yet? Is it possible to eliminate any of the possible final states? Which one(s)? (e) We now need to repeat Part 3b to figure out which happens first: the tea decreases to 0 C, or the ice goes through an entire phase change. Draw NEW initial points and determine which changes in energy you need to compare. Then draw the Energy System Diagrams, and do the calculations. (f) Are the two physical systems in thermal equilibrium at the end of this step? How do you know this? (g) Can you now determine the final state of the ice and tea system? Draw an appropriate Energy-System Diagramon the board for this. If you finish early, you may solve for the final temperature or state. 34

41 Unit II Mechanical Energy Systems 35

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43 Discussion/Lab 6 6 A Modeling with Mechanical Energy Systems Overview: After spending the first few weeks of the semester discussing and making sense of various thermal phenomena, we now shift our focus to mechanical phenomena. We ll start with some new definitions and then immediately dive into some examples. 6 A.1 Internal Energy and Mechanical Energy Please read before moving on: At this point, it is useful to define and distinguish two different types of energy systems. Make sure the definitions make sense to everybody in your group! Internal Energy: Internal energy systems are closely related to the motions of and relationships between individual atoms and molecules. Thermal Energy and Bond Energy are examples of internal energy systems. Mechanical Energy: Mechanical energy systems have indicators involving the position or the speed of the object as a whole. The object can be anything from an airplane to an atom. Important: Some physical phenomena (such as the thermal phenomena we ve dealt with, so far) involve changes only in internal energy systems, while some phenomena involve changes only in mechanical energy systems. Other phenomena involve changes in both mechanical and internal energy systems. 6 A.2 Phenomenon: Dropping a golf ball Drop a golf ball from about one meter above the floor and observe its motion and position. Focus on the motion and position between the beginning and end states defined below. If you do not have names of energy systems associated with the changes you observed, you can just make up names for these energy systems for now. 37

44 DL6 Activity 6 A. Modeling with Mechanical Energy Systems beginning just as ball is released end just before ball hits the floor 1. Write on the board: What properties of the physical system (indicators) do you think change significantly between the initial and final state? What energy systems are associated with those indicators? Are these internal or mechanical energy systems, or both? 2. Put a complete Energy-System Diagram on the board. Show with an arrow if each energy system increases or decreases. Include initial and final values of the indicators. 3. Write an algebraic representation of your Energy-System Diagram that expresses energy conservation in terms of the changes of energy of the relevant energy systems. 6 A.3 Phenomenon: Dropping a coffee filter Drop a basket type coffee filter from a height of about two meters. Release it oriented as shown in the figure. beginning one second after filter is released end just before filter hits the floor 1. Repeat part 1 above. Don t forget the last part of the question! 2. Repeat parts 2 and 3 above. Discuss in your group: 3. Which of the energy systems you have identified depend on only the magnitude of the indicator and which depend on both the magnitude and the sign associated with that indicator? 4. How are your particular models for the golf ball and coffee filter different from each other? Be specific. What physical properties of the objects are responsible for this difference? 38

45 Activity 6 B. A New Construct and A New Model DL6 6 B A New Construct and A New Model Overview: A third mechanical energy system in addition to translational kinetic energy KE translational and gravitational potential energy P E gravity is the spring-mass energy P E spring-mass, introduced here in the context of the Spring-Mass Model. 6 B.1 Oscillating Spring-Mass Many physical systems oscillate. The spring-mass system that you are going to examine is one example of many such systems that can be modeled using the Spring-Mass Model. 1. Hang a 200 g mass on the spring. The equilibrium position of the mass is the location of the mass when the system is at rest. Use the two-meter stick with the pointer to mark the equilibrium position of the mass. Pull the mass down or push it up a few centimeters and release it. Describe the resulting motion of the mass (e.g., Where is it moving fastest? Where is its speed zero?). 2. Does the resulting motion of the mass depend on whether you started the motion by lifting the mass up a certain small distance (try 1 cm) or by pulling it down that same distance from its equilibrium (resting) position? 6 B.2 Applying the Energy-Interaction Model Work out your responses to the questions below with your small group at the board. Remember that in order to identify an energy system, we need to have an indicator that tells us that the energy system is changing. Start with the mass pulled down 5 cm below its equilibrium position. beginning just as mass is released end the instant the mass passes through equilibrium position for the first time 1. What properties of the physical system (indicators) changed significantly between the initial and final state? What energy systems are associated with those indicators? 2. Make a complete Energy-System Diagram: Show the increases and decreases in the energy systems and show the initial and final values of the indicator associated with each energy system be as explicit as possible. 39

46 DL6 Activity 6 B. A New Construct and A New Model 3. Write down an algebraic representation of your Energy-System Diagram, expressing energy conservation. 6 B.3 Reflecting on the Spring-Mass Model 1. Do changes in the energy systems you identified depend on the direction that the indicator changed (the sign of the indicator), or only on the magnitude of the indicator? 2. Would the way you modeled this physical system be any different from what you did in your response to 6 B.2#2 if the final state were the instant the mass returned to its equilibrium position the second time, instead of the first? How about if it were the 200th time? 40

47 Discussion/Lab 7 7 A Creating Particular Models with Mechanical Energies Overview: The problems in this section illustrate how we can use the Energy- Interaction Model to answer questions and make predictions about the allowed behavior of interacting objects that are difficult to approach in any other way. 7 A.1 Three Thrown Rocks FNT 7-1 Before we start with this problem, let s review what the Energy-Interaction Model does for us. As we have said before, conservation of energy relates values of certain physical parameters at the beginning of a process to the values of those parameters at the end of the process. The parameters are typically the indicators of the energy systems that change during the process or interaction. If we have a question about or want to predict values for some parameter and this parameter happens to be an indicator of an energy system, or a coefficient in an expression for an energy system, then we can proceed to construct a particular model and see if it gets us what we want. The Phenomena: Three rocks of equal mass are thrown with identical speeds from the top of the same building. (1) Rock X is thrown vertically downward, (2) Rock Y is thrown vertically upward, and (3) Rock Z is thrown horizontally. The Question: Which rock has the greatest speed just before it hits the ground? Assume air resistance is negligible. How can we determine this? We could take a guess, but it helps our argument if it is possible to apply the Energy-Interaction Model. The prompts below will guide you through the process. 41

48 DL7 Activity 7 A. Creating Particular Models with Mechanical Energies (a) Let s start by making a prediction based on your prior experience. Don t waste a lot of time on this right now, but it s useful to give it some thought. We will definitely come back to this at the end in Part (f). Your first, intuitive ideas might be very useful! (b) Does the question involve a parameter that you know to be an indicator of the change in an energy system? Which energy system and what is the indicator? (c) Rock X: Construct an Energy-System Diagram for Rock X, which is thrown straight down. Write out the expression for energy conservation, based on your Energy- System Diagram, as E s; then substitute in algebraic symbols for the different energy systems. We are not asking to go any further, but if you just have to, go ahead and try solving it for the final speed, v f (but you really don t need to). (d) Rock Y: Now, without actually writing anything down, consider what would be different in your Energy-System Diagram for Rock Y. How about the algebraic representation anything different? Go back and re-read the The Phenomena description at the beginning of the FNT, if you are not sure. Anything different in terms of what goes into the model? In terms of the Energy-Interaction Model are there any differences? Yes or No? Are you 100% sure? Why? (e) Rock Z: Repeat for Rock Z. Any differences in the model? Yes or No? (f) Do you believe that conservation of energy holds true for these three phenomena? Another way to ask this: Does the particular energy model you developed apply to all three cases? If yes, what does conservation of energy tell you about the final speeds of the three rocks with 100% confidence? (g) How do you reconcile your result in Part (f) with your initial ideas from Part (a) above? They are not crazy, because there are indeed very obvious differences in the three situations. Why don t these differences matter to energy conservation? Try to be as explicit here as you can be. This is what we will focus on in the FNT follow-up in DL. Using the prompts below, briefly compare your group members responses to FNT 7-1 and put a short answer on your board (each letter corresponds to the same letter in the FNT): (b) Does the question involve a parameter that you know to be an indicator of the change in an energy system? Which energy system and what is the indicator? (c) Draw an Energy-System Diagram for Rock X with an algebraic representation consisting of two and only two lines: (a) 1 st line, using E s with subscripts. (b) 2 nd line, substitute in algebraic expressions for each E. Do NOT solve for anything! 42

49 Activity 7 A. Creating Particular Models with Mechanical Energies DL7 (d) & (e) Are there ANY differences in the Energy System Diagrams for Rocks X, Y, and Z? (f) (i) In terms of the particular model you have created for these phenomena (determined by which energy systems you included), what does your model predict with 100% certainty about the final speed of each rock just before it hits the ground? (ii) In light of the dropping golf ball and coffee filter activity (Activity 6 A), what aspect of your model might have to be changed that could change your prediction regarding the final speeds of the ball? (iii) So, taking into account what you just determined in the previous two steps, what is different about the final states of the balls? (g) Discuss any intuitive ideas that different members of your group might ve had that bug you the most, because they seem to contradict the Energy-Interaction Model s prediction that all rocks have the same final speed. Check-In: Does this all make sense to you? In the context of this FNT, we discussed some very fundamental issues about energy conservation. If you re not sure about any of the above, please follow up with other students and/or your instructor during office hours. But before you do, carefully work through the entire FNT again! 7 A.2 Pulling up a Bucket FNT 7-2 A person pulls a bucket of water up from a well using a rope. Assume that the initial and final speeds of the bucket are zero (v i = v f = 0), and that the person lifted the bucket a vertical distance h. By looking at energy changes in the bucket-earth physical system, we can make sense of the force the person must exert to pull the bucket up and determine the amount of work the person does. (a) Is the system open or closed? What energy systems undergo a change during this process? Construct an Energy-System Diagram and include the algebraic expression of energy conservation (in terms of E s and, if open, any heat or work). (b) Substitute the algebraic expression we use for the change in gravitational potential energy and solve algebraically for the work done by the person on the rope/bucket. (c) Use the definition of work in terms of force and distance (refer to the Energy- Interaction Model Summary) to find the average force exerted by the person on the rope and bucket while they are lifting it up. 43

50 DL7 Activity 7 A. Creating Particular Models with Mechanical Energies 1. Put on the board the algebraic expression you have for work in terms of energy changes. Be prepared to give an explanation that is both succinct and logically complete based on the Energy-Interaction Model for how you know this expression is true. 2. Substitute the expression for work in terms of force and distance moved (see Energy- Interaction Model Summary), and solve for the force. 7 A.3 Including the Potential Energy of a Spring 1. The Phenomenon A toy car launcher works as follows: There is a compressible spring that is attached to a horizontal rail, on which a toy car can roll. The toy car can be pushed back against the spring, compressing it a certain distance. There is a little hook that holds the car against the compressed spring. When the hook is released, the spring pushes the car out of the launcher, down the horizontal track. We want to ask some questions and make some predictions about different cars as they come out of the launcher. Upon close examination of this launcher, it is noted that the spring is always compressed the same amount each time a car is loaded into the launcher. The spring stays put inside the launcher as the car is pushed out. 2. What we want to know or predict Do all cars come out of the launcher with the same speed? If they don t all have the same speed when they come out, what parameters cause the difference? What do we need to do to get our car to have a higher speed than our friend s cars? 44

51 Activity 7 A. Creating Particular Models with Mechanical Energies DL7 3. Create a Particular Model and Use it to answer the questions Plan what you need to do in your group and how you will proceed BEFORE you start writing on the board. Be systematic. Then put your response on the board. 4. Suggestions: Use the Energy-Interaction Model and the Spring-Mass Model. Consider what energy systems are involved. Note: The car is in contact with the spring while it is compressed and separates from the spring just as the spring reaches its equilibrium length. During the time that the car and spring are together, they behave as a spring-mass system. To simplify the notation, use d for the distance the spring is compressed instead of x. 5. Be prepared to present a precise, logical, explanation that gets at the answers to the questions. 45

52 DL7 Activity 7 B. How We Use the Energy-Interaction Model 7 B How We Use the Energy-Interaction Model Overview: In this activity, we reflect on what we did when using the Energy- Interaction Model. We ll generate some general guidelines for successful application of the model to make sense of physical phenomena. 7 B.1 Reflecting on Using the Energy-Interaction Model In the previous activities we have been applying the Energy-Interaction Model to a wide range of physical phenomena. Previously, we had applied it to thermal and chemical interactions. Here is a list of what we ve just done with mechanical phenomena: Activity 6 A Dropped golf ball and dropped coffee filter Activity 6 B Oscillating mass attached to a spring Activity 7 A Three thrown rocks, Pulling up a bucket, and Toy car launcher In all of these cases we used the Energy-Interaction Model to make sense of the phenomena, answer questions, make predictions, or develop explanations regarding the phenomena. But the Energy-Interaction Model is a very general model that needs to be refined to tell us anything about the particular phenomena. 1. What did you have to do and what decisions did you have to make in order to use/apply the Energy-Interaction Model? There are three major decisions you must initially make that determine how you are modeling the process or phenomenon. The process of making an Energy-System Diagram helps you with this. List the most important three things you can think of here and on your board: (a) (b) (c) 2. Which steps on the two-page blue model summary of the Energy-Interaction Model do the above three steps correspond to? The process of making these decisions/choices is the first step in what we have and will continue to refer to as creating or developing a particular model. This is the model that can be directly applied to a particular phenomenon. 46

53 Activity 7 B. How We Use the Energy-Interaction Model DL7 7 B.2 The Whole Process 1. To complete the creation of the particular model, you must do some more things, perhaps make some decisions. When first encountering a phenomenon, an Energy- System Diagram can be particularly useful because as you draw the diagram, you are forced to make these decisions. 2. Create a particular model for the case of a mass hanging on a spring that is pulled down and released at a distance d below the point where the mass is hanging stationary. After the mass has completed a total of 10 complete oscillations, it is noticed that it didn t go quite as far down as the distance d. Put your particular model, in the form of a complete Energy-System Diagram, on the board. Be prepared to discuss in detail the decisions you had to make when forming this diagram. 3. What are some of the questions that you could ask/answer or predictions you could make with this particular model? You would likely need to have more specific information that you could obtain from the phenomenon. Put these questions on the board with any additional information you would need. 7 B.3 A New Phenomenon Imagine this situation: Two identical balls roll down two sets of slanted channels. The slope and length of the channels are the same, so the change in height is the same for both balls. However, the channels have different widths: one is wider than the other. Your instructor will give you two Pool balls. Let them roll down the channels starting them at the same time and observe what happens. Do this several times, carefully observing what is different about the way the balls roll. Could the difference be an indication that there is a new energy system that we need to take into account? A maybe obvious question is, Why does one ball go faster and get to the bottom before the other ball? Use the Energy-Interaction Model to make sense of what is going on, and develop a succinct explanation with an appropriate Energy-System Diagram for why one of the balls gets to the bottom of the rails before the other ball. Note: From careful measurements made on this apparatus, friction plays only a very small roll in the difference in speeds of the balls. You can safely assume that all frictional effects cause negligible energy changes compared to the changes in mechanical energies. 47

54 DL7 Activity 7 C. Quantitative Modeling of Mechanical Energies 7 C Quantitative Modeling of Mechanical Energies Overview: There are interesting features of some of the Mechanical Energy systems that only become apparent in the algebraic expression for the change in energy. These features can help us tremendously in making sense of physical phenomena. To remind you, the energy systems we are currently working with are KE translational, KE rotational, P E gravity, P E spring-mass, and P E spring. In this activity, we will explore some quantitative features of these energy systems. 7 C.1 Relationships between Energy Systems and Indicators In your small group, take a few minutes to think about and discuss each of the questions below. Each question will be followed immediately by a short 1. What are the indicators of change in each of the Mechanical Energy systems? 2. (a) Which of these energies depend on the square of the indicator and which depend on the first power of the indicator? (b) Which of these energies depend on which direction something moves? (c) What is the connection between the two previous questions? 3. Which of these energy systems depend on the mass of the object and which depend on the weight? Why does this matter? The fact that some indicators are linear and some are quadratic has additional consequences as the next two questions illustrate: 4. Consider a heavy ball. Would it take the same amount of work to vertically lift it from 25 cm to 30 cm as it does to lift it from 30 cm to 35 cm? If not, which situation requires more work? Why? 5. Does it take the same amount of work to speed your car up from 25 m /s to 30 m /s as it does to speed it up from 30 m /s to 35 m /s? If not, situation requires more work? Why? 7 C.2 Keeping Track of Directionality We are about to begin using the Energy-Interaction Model to predict quantitative properties of various phenomena. A potential pitfall of quantitative modeling is that we sometimes tend to lose track of why we are doing something once we start calculating numbers. It is especially easy to lose track of directionality, which is mathematically expressed in algebraic signs. However, our Energy System Diagrams tell us the signs of the various terms in the conservation of energy equation most of the time. Therefore, don t forget to use the Energy System Diagrams to check the signs as you begin to model scenarios quantitatively! 48

55 Activity 7 C. Quantitative Modeling of Mechanical Energies DL7 The Phenomenon: Christine throws a ball straight up, letting go of the ball at a height of y i above the ground. When she lets go, the ball has an initial speed v i. The ball travels straight up to its maximum height (y max ) and falls back down. Assume the frictional effects from air resistance are insignificant. Constructing a particular model to find the maximum height: 1. Construct a particular model of this phenomenon that can be used to determine the maximum height of the ball. (y f = y max ) Put a sketch of the path of the ball and a complete Energy-System Diagram (with accompanying energy conservation equation) on the board [leave half the board free for the following questions]. What do you know about the ball s speed at its maximum height when it s thrown straight up? 2. Indicate the initial and final positions of the ball on your sketch that correspond to the initial and final conditions in your Energy-System Diagram. 3. On the board, rewrite the energy conservation equation by replacing the two terms with algebraic expressions for the changes in energy of the energy systems. Does the resulting algebraic sign of each term in your energy conservation equation agree with the increases and decreases in energy systems in your diagrams? 4. Solve the equation for (y f y i ). Does the sign of (y f y i ) make sense for this particular physical situation? 5. If you knew the numerical value of v i, could you calculate the maximum height? 49

56 DL7 Activity 7 C. Quantitative Modeling of Mechanical Energies 50

57 Discussion/Lab 8 8 More Modeling of Scenarios involving Mechanical Energy Systems Overview: In this activity, we will explore additional scenarios that are related to phenomena we ve already discussed. Here, we add some variations to the scenarios and answer some different questions we haven t asked yet. Note: Remember Christine throwing a ball in the air in Activity 7 C.2? FNT 8-1 If the initial upward speed of the ball in Activity 7 C.2 is 10 m /s, and the ball is released at a height of 1.5 m above the floor, what is the maximum height above the floor that the ball reaches? How far above Christine s hand is the ball when it reaches its maximum height, and how is this value related to your answer to 4 in Activity 7 C.2? FNT 8-2 With the same initial conditions as in FNT 8-1, use the Energy-Interaction Model in two different ways to determine the speed of the ball when it is 4 meters above the floor, headed down: (a) Construct a particular model of the entire physical process, with the initial time when the ball leaves Christine s hand, and the final time when the ball is 4 meters above the floor, headed down. (b) Divide the overall process into two physical processes by constructing two Energy System Diagrams and applying energy conservation for each: one diagram for the interval corresponding to the ball traveling from Christine s hand to the maximum height; and one diagram corresponding to the interval for the ball traveling from the maximum height to 4 meters above the floor, headed down. (c) Did you get different answers in parts (a) and (b) for the speed of the ball when it is 4 meters above the floor, headed down? 51

58 DL8 Activity 8. More Modeling with Mechanical Energy FNT 8-3 Use the Energy-Interaction Model to show that an object thrown vertically upward will have the same speed as it comes down through any point that it had going up through that same point. One way to do this is to construct two Energy System Diagrams: One diagram should be from the point of release of the ball to some intermediate height as the ball is traveling upward, less than the maximum height; the second diagram should be from the point of release of the ball to that same intermediate height as the ball is on its way down. Then, compare the two diagrams. FNT 8-4 In Activity 7 C.2 we assumed that y = 0 at the level of the floor. If, instead, we assume that y = 0 where Christine releases the ball still 1.5 meters above the floor will this change the maximum height above the floor attained by the ball? Use the Energy-Interaction Model to answer this question. After throwing her ball in the air a few times, Christine is tired of playing with the ball and instead moves on to water balloons (much more fun, right?). She stands on top of the Science Building, ready to launch her water balloon. FNT 8-5 Christine drops a water balloon from the top of the Science Building. Let s assume that the balloon does not break when it strikes the ground. There are many questions we could ask about this situation. To answer any of them, it makes sense to model the Energy dynamics of the scenario first. Let s do that and then answer some particular questions! (a) Create a particular model for this phenomenon by making an Energy-System Diagram for the process that takes place from the time the water balloon is dropped until it is motionless on the ground. Consider the indicators to determine what energy systems must be present/can be excluded. Have you included enough systems? (b) If the water balloon falls a distance of 21 m, what is the maximum temperature rise of the water balloon due to its being dropped? (Does you answer seem reasonable? Why or why not? It may help to check your units.) (c) Is there anything that prohibits the water balloon from suddenly cooling off to its original temperature and leaping 21 m into the air? From the random nature of thermal energy, why do you think we never see this happen? Respond briefly. Eventually, Christine grows tired of throwing and dropping things. She goes back into the Science Building to play with a mass that s hanging on a spring in one of the laboratory rooms. 52

59 Activity 8. More Modeling with Mechanical Energy DL8 FNT 8-6 Christine s mass-spring system consists of a 250 g mass hanging from a spring with a spring constant k = J /m 2. She pulls the mass down 7.1 cm from its equilibrium position and then releases it from rest. (a) How much work did Christine do when she pulled the spring down from its equilibrium position? Assume that the mass was at rest before she pulled it down, and before it was released. (Use the Energy-Interaction Model, not the expression W = F avg x, to determine the work.) (b) Create a particular model (construct an Energy-System Diagram) for each of the following final conditions to predict the speed of the mass after it is released, and when it is: (i) moving up through the equilibrium position, (ii) moving down through equilibrium, and (iii) 5.0 cm below the equilibrium position, moving down. Each group is responsible for putting one of the following on the board. Write so that all other groups can easily follow your presentation! Group 1 FNT 8-1 & FNT 8-4 (<5 min) Read: You should already have used the principle of energy conservation to develop an equation for the change in height of a ball thrown straight up in the air (from Activity 7 C.2). Do: Write a sentence or two explaining what fundamental feature of the Energy-Interaction Model accounts for why changing the location of y = 0 does not change your calculation of the ball s maximum height above the floor. [Hint: think about the general form of the algebraic expression of energy conservation.] FNT 8-2 (<5 min) Read: When you model the process described in FNT 8-2 in two different ways, i.e., using an interval corresponding to the overall process in versus splitting the overall process into two contiguous pieces with separate intervals corresponding to each part, you should get the same result for the speed of the ball at 4 m above the floor. Check with your instructor for the correct numerical answer. Do: In terms of the Energy-Interaction Model, why is it not necessary to divide the overall process into two pieces in order to find the speed of the ball at 4 m above the floor falling down? 53

60 DL8 Activity 8. More Modeling with Mechanical Energy Group 2 FNT 8-3 (<10 min) Construct one or two Energy System Diagrams that you can use to explain how you know that an object thrown vertically upward will have the same speed as it comes down through any point that it had going up through that same point. Group 3 FNT 8-5 (<10 min) Construct a complete Energy-System Diagram for the process described in Part (a) of this FNT. Group 4 FNT 8-6, Part (a) (<10 min) Construct a complete Energy-System Diagram for the process described in Part (a) of this FNT. Group 5 FNT 8-6, Parts (b)-(i) & (b)-(ii) (<10 min) Construct a complete Energy-System Diagram for the process described in Parts (b)-(i) and (b)-(ii) of this FNT. Group 6 FNT 8-6, Part (b)-(iii) (<10 min) Construct a complete Energy-System Diagram for the process described in Part (b)-(iii) of this FNT. 54

61 Discussion/Lab 9 9 A Graphically Representing Energy Relationships Overview: By now, you are very familiar with an algebraic representation of the energy conservation principle. In this activity, we are restating this algebraic representation and introducing a graphical representation of this principle as a useful tool for understanding certain physical phenomena, for example a falling ball. 9 A.1 Rethinking and Restating Energy Conservation Situation 1: A dropped ball at any time between when it is dropped from rest to just before it hits ground. Our standard expression of energy conservation for this situation is P E gravity + KE = 0. However, there are times when it is more useful to have a different expression of energy conservation (e.g., when graphically representing energy amounts). Your instructor will use the definition of the difference operator to algebraically show how the expression: is equivalent to: E total (at any time) = const. = P E gravity + KE P E gravity + KE = 0. 9 A.2 Using Energy Conservation to Graphically Represent Energy Amounts for the Falling Ball In order to plot the different amounts of energy (P E gravity, KE, and E total ) of the dropped ball (Situation 1 above) as a function of height, it is necessary to decide where to set y = 0. This means, we need to decide where in space to put the origin of the y-coordinate, which measures the height of the ball above the surface of the earth. For example, you could choose y = 0 to be at the floor level, or at 3 m below the floor. Since we use P E gravity = mgy as our standard expression for gravitational potential energy, the place where P E gravity = 0 depends on where we put y = 0. For any given physical situation you can set y = 0 wherever you want. 55

62 DL9 Activity 9 A. Graphically Representing Energy Relationships Your instructor will demonstrate how to create the graphical representation of energy amounts with y = 0 set at 3 m below the floor. Then, each small group will construct an energy graph for a dropped ball following the directions given below for where to set the origin of the y-coordinate system. In your group: Assume the ball is being dropped from a height of 2 meters above the floor (Note: This physical situation is exactly the same for each group). Graph the three energy amounts in Situation 1 using the y = 0 location listed here: Group y = 0 at: Group y = 0 at: 1 2 m below the floor 4 2 m above the floor 2 1 m below the floor 5 3 m above the floor 3 1 m above the floor 6 4 m above the floor (a) First, create a small, simple sketch of the physical scenario, indicating where y = 0, the ball s initial position, and the floor. Label each of these places with its respective y-value and a physical description. (b) Plot P E gravity, E total, and KE of the ball (all on the same graph) as functions of height using coordinate axes as described above. Why does it make sense to graph P E gravity first, E total second, and only then KE? Be ready to explain how you determined E total. 1 Be prepared to explain how you constructed your graph. (c) Choose an arbitrary value of y on your graph. For this value, indicate on your graph how E total = P E gravity + KE. Then, choose two different values of y on your graph. For those values, indicate on your graph how P E gravity + KE = 0. 9 A.3 Using Energy Conservation to Graphically Represent Energy Amounts for a Mass-Spring System Situation 2: A mass hanging on a spring is pulled down and released. In your plot, consider all times between when the mass-spring is released and the first time it is momentarily at rest. Repeat (a), (b), and (c) from Part 9 A.2 using Situation 2. The three energy amounts are now plotted as functions of distance from equilibrium. In this situation, everyone should have y = 0 at the same place: The equilibrium position of the mass-spring system. 1 Remember: It is standard convention to plot the dependent variable on the vertical axis and the independent variable on the horizontal axis. Consequently, the energy is plotted on the vertical axis. 56

63 Activity 9 B. Explicit Reasoning Using Models DL9 9 B Explicit Reasoning Using Models Overview: Now that we ve discussed many different physical scenarios, our models probably have become second nature to you. Maybe so much so that we may have to revisit how to explicitly model a phenomenon. We ll do that with a new phenomenon. 9 B.1 Two Masses Over a Pulley Atwood s Machine 1. Think and Discuss Imagine this situation: A string connecting a smaller mass, m, and a larger mass, M, passes over a small pulley that is almost frictionless. The masses are initially held at rest; then are released and allowed to move: Based on your intuition which set of attached masses, each differing by 20 g, will move faster when released? Set A: M A = 220 g, m A = 200 g Set B: M B = 70 g, m B = 50 g Circle your choice here: (a) Set A moves faster (b) Set B moves faster (c) Both move with the same speed 2. Put your group s choice on the board. How sure are you about your reason(s) and predictions? What is your intuition based on? 3. Try it out Pair up with an adjacent table. One table will attach the heavier set of masses, the other table should attach the lighter set. Try it out and see which set moves faster. What do you observe? 57

64 DL9 Activity 9 B. Explicit Reasoning Using Models 9 B.2 Applying the Energy-Interaction Model Apply the Energy-Interaction Model to the two-masses-over-a-pulley situation. Model this system as if the pulley is massless and frictionless, so you won t have to worry about energy systems associated with the pulley. Take the initial state to be just as you release the masses (what is v i? what is v?). Take the final state to be when the masses have moved a distance d and have speed v, but before they hit anything or run out of string Create a particular model of the phenomenon described above by constructing a complete Energy-System Diagram for each of the mass sets using the initial and final states described above. 2. When you substitute algebraic expressions for changes in individual energy systems in the algebraic representations of your particular Energy-Interaction Models, you will find the symbols m, M, g, v, and d useful. Watch your minus signs! 3. One important practical use of the Energy-System Diagram is in the interpretation of algebraic expressing of energy conservation. For example, what do each of the terms in the equation mean, and what should their sign be? Check, and be ready to show, that the signs of the terms in the equation from (2) are consistent with the increases and decreases in energy of the energy systems in your diagram. 4. What energy systems have you ignored by your assumptions about the pulley? [You don t have to put (4) on the board.] 9 B.3 Reasoning and Explaining with Particular Models You have constructed two particular Energy-Interaction Models, one for each of the different sets of masses. You are now going to use these models to make sense of and explain why one set of the masses (the set with the smaller masses) moved faster than the other set. Comparing the changes in energy of the various energy systems 1. During the process (from beginning to end), does the total KE of the two masses (the sum of the KEs) increase, decrease, or stay the same? How do you know? Is your response the same for both the heavier and the lighter pair of masses? 2. During the process, does the total P E of both masses (sum of the P Es) increase, decrease, or stay the same? How do you know? Is your response the same for both the heavier and the lighter pair of masses? 2 Since d denotes a distance, it is a positive number; so, y = ±d, as appropriate. 58

65 Activity 9 B. Explicit Reasoning Using Models DL9 3. Compare the magnitudes of the total change in P E of both masses that occurs during the process across the two systems (the heavier pair and the lighter pair). Are the total changes the same or different? Explain. 4. Compare the magnitudes of the total change in KE of both masses that occurs during the process across the two systems (the heavier pair and the lighter pair). Are the total changes the same or different? Explain. 5. Did the original question regarding the two sets of masses have to do with the KE of the masses or something else? Was it related to the KE? Compare this across the two systems (the heavy pair and the lighter pair). Is this the same or different? Explain. 9 B.4 Creating a Convincing Scientific Explanation Turn your responses to the questions (1) through (5) from 9 B.3 into a set of statements that constitute a logical argument based on your particular Energy-Interaction Models that explains why the set of masses with smaller total mass move faster than the set with greater total mass, as long as the difference between the two masses in each set is the same. 59

66 DL9 Activity 9 B. Explicit Reasoning Using Models 60

67 Discussion/Lab A Practicing Graphical and Quantitative Modeling of Mechanical Energy Systems Overview: Before we introduce a new class of physical phenomena, forces, we want to make sure you get a bit more practice modeling phenomena using the tools we have accumulated until now, especially the method of graphing energy amounts introduced in the last activities and the carefully crafted model-based, scientific arguments we ve been practicing for a while. We ve done a practice activity like this before: At home, please work through all the FNTs listed in this activity. In class, each group will work on a portion of these FNTs to come to a consensus before we discuss the scenarios with the whole class. FNT 10-1 This FNT is excellent practice in constructing a scientific argument based on logic that follows directly from the physical situation and the relationships of the models you use to, well, model the situation. This is what science is all about! Please refer back to Activity 9 B: (a) Finish any of the prompts from 9 B.2 and 9 B.3 that you were unable to finish during the last discussion lab. (b) Respond to the prompt in 9 B.4, which asks you to turn your responses to prompts from 9 B.2 and 9 B.3 into a set of statements that constitute a logical argument based on the Energy-Interaction Model. With your argument, explain why the set of masses with smaller total mass moves faster than the set with greater total mass as long as the difference between the two masses in each set is the same. 61

68 DL10 Activity 10 A. More Modeling of Mechanical Energy Systems FNT 10-2 A 0.4 kg mass is attached to a spring that can compress as well as stretch (spring constant 50 N /m). The mass and spring are resting on a horizontal tabletop. The mass is pulled, stretching the spring 48 cm. When it is released, the system begins to oscillate. (a) Assuming the transfer of energy to thermal energy systems is negligible, construct a complete Energy-System Diagram that could be used to predict the speed of the mass as it passes a point that is a distance of 39 cm from its equilibrium point on the other side of the equilibrium position (spring is compressed). Substitute all known values of constants and variables into the algebraic expression of energy conservation, and identify any unknown(s). Do you have enough information to find the speed of the mass? (b) Now assume that the effects of friction are not negligible. When pulled back and released as before, the mass now reaches its furthest distance from equilibrium at 40 cm on the compressed side (before bouncing back again). Construct a complete Energy-System Diagram that could be used to determine the amount of energy transferred to thermal systems when going from the initial stretched position to where it first momentarily stops. Proceed as in part a: Substitute all known values and identify any unknown(s). Can you determine the increase in thermal energy? FNT 10-3 A skier (of mass 55 kg) skies down the smooth (frictionless) ski slope illustrated in the cross-sectional diagram. She pushes off at the top with a speed of 10 m /s. At the bottom (0 m), she comes to a stop by digging her skis in sideways. (a) Construct a complete Energy-System Diagram that can be used to predict the speed of the skier when she is on the middle flat part (at 10 m). Substitute all known values of constants and variables, and identify any unknown(s). Do you have enough information to determine the speed at this part of the hill? (b) Construct a complete Energy-System Diagram that can be used to predict the skier s maximum speed just before digging in her skis at the bottom. Substitute for constants and variables and identify any unknown(s) as in Part (a). (c) Assuming that the snow at the point where she comes to a stop is at a temperature of 0 C and that all of the kinetic energy of the skier goes into melting the snow, construct a complete Energy-System Diagram that can be used to predict the amount of snow melted by the skier while stopping. Substitute for constants and variables and identify any unknown(s) as in Part (a). 62

69 Activity 10 A. More Modeling of Mechanical Energy Systems DL10 FNT 10-4 Christine (from Activity 7 C.2) throws a 300 g ball straight up into the air. The ball is exactly 2 m above the ground when Christine lets go of it. It reaches a height 12 m above the height from which it was released, and then falls straight back down. (a) Assume that y = 0 at the point of release of the ball. minimum values of y and P E gravity. Find the maximum and What is the total energy of the system? Find the maximum and minimum values of KE. (b) Repeat Part (a), with y = 0 at the ground. FNT 10-5 A 200 g mass is attached to a spring, just like in Activity 9 A.3. The mass is lifted up 5 cm and released so that it begins to oscillate about the equilibrium point. The spring has a spring constant k = 500 N /m (= 500 J /m 2 ). (a) Calculate and accurately plot on a letter size ( in) sheet of graph paper P E spring-mass, E total, and KE. The vertical axis of the graph should be energy (in Joules). The horizontal axis is distance from equilibrium (in Meters). (b) On the same graph, quickly sketch (without calculating values) the P E spring-mass, KE and E total of the system if the mass were initially pulled back (stretched) 2.5 cm from its equilibrium point, instead of lifted up (compressed) 5 cm. FNT 10-6 Remember the physical situation described in FNT 10-3? To refresh your memory: A skier (of mass 55 kg) skies down the smooth (frictionless) ski slope illustrated in the cross-sectional diagram. She pushes off at the top with a speed of 10 m /s. At the bottom (0 m), she comes to a stop by digging her skis in sideways. Use the algebraic expression of energy conservation that shows that the sum of all the energies at all points in time is constant and equal to the total energy for the following: (a) Pick a location to set y = 0 and determine the total energy of the system. (b) Write an algebraic representation expressing energy conservation that you could use to find the speed of the skier when she is on the middle flat part (at the height 10 m). (c) Substitute all known values of constants and variables, and identify any unknown(s). 63

70 DL10 Activity 10 A. More Modeling of Mechanical Energy Systems Each group is responsible for putting one of the following on the board. Write so that all other groups can easily follow your presentation! Group 1 FNT 10-4 For each of Parts (a) and (b) of this FNT, make a simple sketch (not a graph) showing the origin (where y = 0), the ball s initial position, positions y max and y min, and the floor. Then, for each of your two sketches, show how to find the maximum and minimum values of gravitational potential energy, and the total energy. Be ready to explain how changing the location of where y = 0 will affect P E gravity. Group 2 FNT 10-5 (a) Draw P E spring-mass and E total (but not KE) on a graph of Energy vs. Distance (x) from equilibrium for the case of the mass displaced 5 cm from equilibrium and released. Clearly label the maximum and minimum values of P E spring-mass. (b) Demonstrate how to use point by point plotting to graph the KE by doing the following: For at least two different values of x, use energy conservation in the form: KE = E total P E to plot a point representing the KE. Label the graph clearly showing P E, E total, and KE at each point. In the you can show how to then fill in the KE by sketching a dashed line connecting the points with a smooth curve. (c) On the same graph, quickly draw P E spring-mass and E total (but not KE) for the case of the mass displaced 2.5 cm from equilibrium. Group 3 FNT 10-2 Part a Construct a complete Energy-System Diagram with accompanying equations as directed in Part (a) of this FNT, and be ready to explain if you have enough information to determine the speed of the mass. FNT 10-2 Part b Construct a complete Energy-System Diagram with accompanying equations as directed in Part (b) of this FNT, and be ready to explain if you have enough information to determine the increase in thermal energy. 64

71 Activity 10 A. More Modeling of Mechanical Energy Systems DL10 Group 4 FNT 10-3 Part a Construct a complete Energy-System Diagram with accompanying equations as directed in Part a of this FNT, and be ready to explain if you have enough information to determine the speed of the skier on the middle part of the hill. Group 5 FNT 10-3 Part b Construct a complete Energy-System Diagram with accompanying equations as directed in Part (b) of this FNT, and be ready to explain if you have enough information to determine the speed of the skier at the bottom of the hill. Could you have used different initial values of energy system indicators to answer the same question? Briefly show/explain on the board. Group 6 FNT 10-6 Put your response for this FNT on the board. Be sure to show how you determined the total energy of the physical system. Is this the only correct value of total energy for the system? Briefly show/explain. All Groups FNT 10-1 Put your response for this FNT on the board. 65

72 DL10 Activity 10 B. Connecting Potential Energy with Forces 10 B Connecting Potential Energy with Forces Overview: We have studied two different potential energy systems so far, P E gravity and P E mass-spring. Our goal in this section is to understand how the forces associated with these potential energy systems are related to the graph of Potential Energy as a function of position. 10 B.1 Graphs for Gravitational Potential Energy 1. Sketch a graph of P E gravity vs. the vertical position y of a ball that is thrown upward from the surface of the Earth. Let y = 0 be at the ground. Note: Be sure to plot P E on the vertical axis and y on the horizontal axis. 2. Next to your graph, write a mathematical expression for P E gravity as a function of y. 3. Consider a particular value of y that is greater than y min and less than y max. At your value of y, what is the direction of the force of gravity on the ball (in the direction of decreasing y or increasing y)? Put an arrow on your graph showing the direction of the force. Repeat for a different value of y. How do the magnitudes of the forces at these two points compare? 4. What, in general, can you say about the magnitude and direction of the force at different y-values? 10 B.2 Graphs for Mass-Spring Potential Energy 1. Sketch a graph of P E mass-spring vs. the distance x of a mass from its equilibrium position in an oscillating mass-spring system. Remember that for a mass-spring system, we always set the origin of the coordinate system at the equilibrium point. 2. Next to your graph, write an algebraic expression for P E mass-spring as a function of x. 3. Consider a particular value of x that is greater than zero and less than x max. At your value of x, what is the direction of the force on the mass (away from equilibrium or toward equilibrium), and is that in the direction of increasing P E or decreasing P E? Note: You can use the mass-spring system at your table to test this. Pull the mass down 2 cm from equilibrium, and then push it up 2 cm from equilibrium. Put an arrow on your graph showing the direction of the force. Repeat for a different value of x. How do the magnitudes of the forces at the two points compare? 4. What, in general, can you say about the magnitude and direction of the force at different x-values? 66

73 Activity 10 B. Connecting Potential Energy with Forces DL10 10 B.3 The Big Idea Remember our goal: We want to relate our observations about the gravitational forces in 10 B.1 and the mass-spring forces in 10 B.2 to features of the graphs of the potential energy systems associated with these forces. That is, we want to find a relationship that holds true for both forces. This relationship will need to relate both the magnitude of the force and the direction of the force to specific features of the graphs of the respective potential energies. The two features of the graphs we will focus on are the instantaneous slope of the graph of P E(y) or P E(x), and the change in P E as the position changes. 1. Can you find any correlation between the magnitude of the force and the magnitude of the instantaneous slope of the P E for these two forces? 2. Can you find any correlation between the direction of the force and the direction of decreasing P E for these two forces? 3. Come to a group consensus about how to state this relationship most succinctly and clearly in words. Write this statement on the board. Physicality Check When you use the feature you found above to predict a force from the potential energy vs. distance graphs, do your results make sense physically? For example, do the directions and magnitudes of the forces found at various points on the graph agree with what you know about the force? Try this with various points on both sets of graphs. Be ready to demonstrate this for the whole class. 67

74 DL10 Activity 10 B. Connecting Potential Energy with Forces Follow-up of Module 2.4 FNTs Groups 1 & 2 discuss and respond to FNT as directed below. Groups 3 & 4 discuss and respond to FNT as directed below. Groups 5 & 6 discuss and respond to FNT as directed below. FNT (<10 min) ) Come to a consensus in your SG about how the changes in P E grav compare for the two situations. ) Come to a consensus in your SG about how the maximum heights compare. All members of your group must now go to the board and work on this together ) After you reach consensus (and check with your instructor) about () and (), sketch the two plots of P E gravitational on the same graph of Energy vs. Height. ) You know that the force of gravity is constant near the surface of the earth and near the surface of the moon, and that it is approximately six times stronger for the earth than for the moon. Make sure everyone in your SG is ready to explain how your graph conveys this information. All members of your group must now go to the board and work on this together FNT (<10 min) ) For two masses connected by a spring the potential energy is given by: P E spring = 1k(r r 2 0) 2 How does doubling the force constant affect P E spring at any particular value of r? ) On one graph, sketch two plots of P E for the two different springs. Pick some particular value of r, and show explicitly on your graph how doubling the spring constant affects the potential energy of the system. How do these two systems differ physically? ) On a second graph, sketch two plots of P E for the two different values of r 0. How do these two systems differ physically? All members of your group must now go to the board and work on this together FNT (<10 min) ) Why do we define the P E between atoms or molecules to be zero when r is very large (i.e., when the atoms are far apart instead of when they are close together)? ) Sketch three separate graphs of P E vs. separation distance on the board: one for a pair of positively charged particles, one for a pair of negatively charged particles, and one for two particles with opposite charges. Make sure everyone in your SG is ready to explain why the graphs are drawn as they are. 68

75 Unit III Forces and Motion: The Momentum Conservation Model 69

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77 Discussion/Lab A Adding and Subtracting Vectors Overview: After studying the energy dynamics of mechanical systems, we re now moving on to take a look at some of the underlying mechanisms and some of the effects of energy transfers and transformations: forces and motion. Since we will spend some time and effort on quantitatively describing both, we need to get familiar with mathematical entities that will make this quantitative treatment possible: vectors. 11 A.1 Vectors Often, a physical quantity cannot be fully described without giving both a magnitude and a direction for it. For instance, consider two physical quantities, mass and force. Suppose I have an object with a mass of 5.0 kg and another with a mass of 1.5 kg. What is the range of possible values for the total mass of these two objects (i.e. how big might the total mass be and how small might it be)? Now suppose I push on some object with a force of 2.5 N and you push on it with a force of 1.5 N. What is the range of values for the total force on the object? Explain. (Hint: the forces might be in the same directions or in opposite directions.) 11 A.2 Vector Addition 1. The picture to the right shows two force vectors. F1 represents a force of 2.5 N pushing straight up the page and F 2 a force of 1.5 N pushing straight up the page. Decide in your groups what the magnitude and direction of the total force acting on an object would be if both F 1 and F 2 were acting on it at the same time. Then decide how you should arrange the two vectors that are shown to represent the addition of these vectors showing how you get the total vector that you expect. Hint: You can easily move the vectors around so that either: (i) their tails touch, (ii) their heads touch, or (iii) the tail of one touches the head of the other. 71 F 1 F2

78 DL11 Activity 11 A. Adding and Subtracting Vectors Which of these three methods best demonstrates addition of these two vectors to find the proper total vector? Sketch this method on the board and also sketch the total force (also called the net force or the sum of the forces, F net = F ). 2. The picture to the right shows three force vectors. Show that your method from Part 1 gives a sensible result if F 1 and F 2 are the two forces acting on an object. Show that your method works if F 1 and F 3 are the two forces acting on an object. F 1 F2 F3 11 A.3 Vector Subtraction 1. An object starts at the initial location (x i, y i ) as shown y in the x-y graph to the right and then follows the dashed line path shown to a final location (x f,y f ). The picture also shows the initial and final position vectors (position is always measured with respect to some axis origin so all position vectors are drawn from the origin to the position of the object). Draw this figure on the board. Then decide in your groups what arrow you would draw to represent the change in the position, R (both the distance moved as well as the direction moved) and then draw this vector on the board. R i R f (x i, y i ) (x f, y f ) 2. Like before, we define the change in a quantity to be the final value minus the initial value: R = R f R i. Decide in your groups whether R i and R f should be arranged tail-to-head, tail-to-tail, etc. to represent the subtraction of these vectors and show how you get the change R that you expected. (Hint: This might be easier thank you think.) 3. Does the method used above give you a reasonable v for the two velocity vectors shown below? x v i v f 72

79 Activity 11 B. Working with Position and Displacement Vectors DL11 11 B Working with Position and Displacement Vectors Overview: Now that we know the basic properties of vectors, let s talk about two similar, yet very different kinds of vectors: position and displacement vectors. Phenomenon: You are in a strange city that has streets that are laid out in a perfectly square grid. Your job will be to move around to different locations as instructed and record your progress. Make sure everyone in your group fully understands the ideas behind each question or part in these activities before going on to the next part. 1. Make a drawing on the board showing the streets in the central city (this should be a square grid with at least 10 streets going in each direction). Decide as a group how you are going to label the streets and use your labeling system. Make sure your diagram is large enough for everyone in the room to clearly see it. 2. Near one of the edges of the diagram, label a street corner o for origin of your coordinate system. Your origin should be different from the origins in other groups drawings. Sketch a coordinate system so that the y-axis points North and the x-axis points East. You start walking somewhere in the city, so choose an intersection near the lower middle part of your diagram, and call that location your initial position. Draw a position vector on your diagram that shows this initial position. Label this vector R i. (a) Write R i, in terms of its x- and y-components, as (R i,x, R i,y ). Determine the length of R i. What units does it have? How would you describe the direction of the position vector? (b) Starting from your initial location, imagine walking a distance equal to 4 blocks North and then 1 block West and then 1 block South to a final location. Show the position vector, R f, for this new location and write R f in terms of its components as (R f,x, R f,y ). (c) As you might expect, we define the change in position to be R = R f R i, so that R i + R = R f. Using the tail-to-head method shown on Page 38 of the course notes, redraw the vectors R i and R f off to the side of your map. Then show how you can graphically obtain R from R i and R f. Make sure you draw these vectors with the same lengths and directions that they have on your map. Using the same picture that you used to show R = R f R i, show that R i + R = R f is also true. Now transfer your R over to your street diagram. 73

80 DL11 Activity 11 B. Working with Position and Displacement Vectors (d) In physics, we are interested in describing motion. If you could choose only one vector from R i, Rf, and R to describe your motion, which one would it be? Why? What do either of the other two vectors, by themselves, tell you about your motion? (e) How are the x-components of R i and R f related to the x-component of R? How about the y-components? (f) Suppose you walked the 4 blocks North and 1 block West and 1 block South in 60 seconds. Draw the average velocity vector for this situation. Write this average velocity vector in component form, (v x, v y ). What is the length of this vector? Include the units. Draw another velocity vector for a situation where you took 180 seconds for this 6 block walk. Which arrow is longer? Why? (g) In which direction(s) would your instantaneous velocity be? (h) If you now walked back to your initial point and assuming the total journey took eight hours what is your average velocity for the entire trip? 3. Look around the room at the different street diagrams. What is similar in each one? What is different? Discuss in your group how you would summarize the meaning of everything you did in this activity. Be prepared to share with the whole class. 74

81 Discussion/Lab A Check for Understanding: Vector Operations Overview: In this section, we ll apply our newly-gained knowledge about vectors. FNT 12-1 Three force vectors are added together. One has a magnitude of 9 N, the second one a magnitude of 18 N, and the third a magnitude of 15 N. What can we conclude about the magnitude of the net force vector? Explain. I. It must equal 42 N. II. It cannot be 0 N. III. Anything from -42 N to +42 N. IV. It can be anything from 0 N to 42 N. V. None of the above can be concluded. FNT 12-2 Alice, Bob and Chuck are three friends standing around, talking. We know that Alice is standing 9 m away from Bob, and that Bob is standing 3 m away from Chuck. Let R AB be the vector that starts at Alice and ends at Bob, and R BC be the vector that starts at Bob and ends at Chuck. (a) Write an equation to find the displacement between Alice and Chuck. (b) What is the furthest distance Chuck can be from Alice, given the information in the question? What is the closest they can be? (c) Draw a picture where Chuck is neither as close as he can be or as far as he can be from Alice. Draw and label the vectors R AB and R BC. 75

82 DL12 Activity 12 A. Check for Understanding: Vector Operations FNT 12-3 Using a piece of graph paper, carry out the operations listed below on the vectors shown at right. Label all vectors. A (a) A + B (b) B + A B (c) A - B C (d) A + B + C (e) B - A (f) - C FNT 12-4 Vectors F 1 on object, F 2 on object, and F 3 on object are all exerted on an object, adding together to form a net force vector, ΣF, as shown in the graph to the right. However, only vectors F 1 on object, F2 on object, and ΣF are known. On a separate piece of graph paper, use the properties of vector addition to graphically determine the vector F 3 on object. Σ F = F 1 + F 2 + F 3 F 1 on object object F 2 on object 76

83 Activity 12 B. Representing the Motion of a Mass along a Circle DL12 12 B Using Vectors to Represent the Motion of a Mass Moving in a Circle Overview: So far, we ve only talked about linear motion an object moving along a straight path. However, many phenomena have objects moving along curved paths. As an example, we ll take a closer look at an object moving along a circle. Phenomenon: You are going to swing a ball in a circle and represent its velocity in several different ways using vectors. Make sure everyone in your group fully understands the ideas behind each question or part in these activities before going on to the next part. 1. (a) Get the ball swinging in a large horizontal circle, going clockwise when viewed from above: Imagine looking down on the ball. Pick a position in space that you will identify as the 12 o clock position. Draw the circular path of the ball on the board with the 12 o clock position toward the top of the board. Choose a coordinate system and sketch it on the board. (b) Draw a position vector identifying the position of the ball when it is in the 4 o clock position. Show the x- and y-components of this vector on your diagram. Discuss in your group how to do this. Be prepared to share with the whole class. (c) Draw another position vector when the ball is in the 5 o clock position. To the side of your diagram, graphically subtract the position vectors as accurately as possible to find the displacement vector; label it r. Describe in a sentence what the delta means here. Brief 2. (a) Describe in words the velocity of the ball as it moves in its circle. Be prepared to share. 77

84 DL12 Activity 12 B. Representing the Motion of a Mass along a Circle (b) What do you think is an instantaneous velocity vector? How might it be different from a position vector? Does it always point in the same direction? If you are having trouble answering these questions, apply them to a specific situation (e.g. driving northwest on I-5 at 65 mph and then curving toward the north). 3. (a) Imagine that you are sitting on the ball and driving it in a circle. Which way are you moving when you are at the 4 o clock position? Draw a velocity vector (put the tail of the vector at the ball s location) showing the velocity of the ball at the 4 o clock position. Show the x- and y-components of this velocity vector on your diagram. Discuss in your group how you do this. (b) Looking at the vectors you have drawn, is a position vector or a displacement vector more closely related to a velocity vector? Make sure this agrees with your knowledge of the definition of velocity. (c) Which vectors on your diagram would be different if you changed the origin? 4. Draw the velocity vector for the ball at the 6 o clock position. Off to the side, graphically determine the change in velocity, v, between the 4 o clock position and the 6 o clock position. 5. (a) Find the x- and y-components of the 6 o clock velocity vector. (b) Find the x- and y-components of the vector v. 6. How can you use the x- and y-components of the 4 o clock and the 6 o clock positions to get the change in velocity, v? Summarize your method and be prepared to share it with the class. 78

85 Discussion/Lab A Force Model: Net Force and Force Diagrams 1 Overview: Now that we can represent linear and curved motion with vectors, let s see how vectors are tremendously useful to describe forces, as well. Phenomenon: You are going to pull on a metal ring with three ropes attached. A spring is attached to each rope to determine the tension in the rope. Before you start, make sure each scale is set to zero by twisting the knob. Work together as a group. No, really: This takes multiple hands! Attach the ropes with the scales turned so you can read force values in Newtons. Note: To get the angles to be 130, trace the ring on a sheet of paper and draw a vertical line pointing down from it. Use a protractor to mark the 130 angles from the vertical line. Line up the ropes along the lines you drew, pull and then note the force scale readings. F Scale 2 on Ring F Scale 1 on Ring Ring F Scale 3 on Ring Scenario 1: Person 1 and Person 2 pull directly opposite to Person 3 so that the yellow spring scale for the third person reads 30 N. Record the values on all three scales. Scenario 2: Persons 1 and 2 change the direction they are pulling until their ropes make an angle of about 130 with respect to Person 3. Record the readings of all scales. 1 See Pages of course notes and Force Model Summary. 79